3rd International Conference

GEOMETRY AND TOPOLOGY OF MANIFOLDS

Krynica, 29 April – 5 May, 2001

Published by INSTITUTE^{OF} MATHEMATICS ^{of} the Technical University of ŁÓDŹ Contents

Foreword 3

Organizers 4

List of participants 5

Titles of lectures 9

Abstracts. Papers 11 Ivan BELKO, The Structure Invariants of a Transverse G-Structure . . . . . . . 12 Moulay–Tahar BENAMEUR and Victor NISTOR, Homology of Complete Sym-

bols and Non-commutative Geometry . . . . . . . . . . . . . . . . . . . . . . 14 Andrzej BOROWIEC, Metric-polynomial structures from gravitational Lagrangians 15 Jaime CAMACARO, Application of Lie algebroids on field theories . . . . . . . 16 Stanisław EWERT-KRZEMIENIEWSKI, On a class of Ricci-recurrent manifolds 17 Vasyl FEDORCHUK, On invariants of continuous subgroups of the generalized

Poincaré group P (1, 4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Rui Loja FERNANDES, Integrating Lie algebroids to Lie groupoids . . . . . . . 22 Janusz GRABOWSKI, Jacobi structures revisited . . . . . . . . . . . . . . . . . . 23 Roman KADOBIANSKI and Vitaly KUSHNIREVITCH, Differential Geometry

Structure for Formal Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Oldřich KOWALSKI, Generalized Symmetric Spaces . . . . . . . . . . . . . . . . 25 Jan KUBARSKI, Cohomology of flat connections in some Lie algebroids . . . . . 36 Vitaly KUSHNIREVITCH and Roman KADOBIANSKI, Configuration Spaces

and Algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Andrzej Krzysztof KWAŚNIEWSKI, Extended finite operator calculus – an ex-

ample of algebraization of analysis . . . . . . . . . . . . . . . . . . . . . . . 47 V. P. MASLOV and Alexandr MISHCHENKO, Geometry of Lagrangian mani-

folds in Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Piotr MORMUL, Geometry of Goursat flags and their singularities of codimension 2 66 Michel NGUIFFO BOYOM, KV-cohomology of contact manifolds . . . . . . . . 73 Andrzej PIA̧TKOWSKI, On prefoliations of the K- space . . . . . . . . . . . . . 74 Paul POPESCU, Submodules of vector fields and algebroids . . . . . . . . . . . . 78 Paul POPESCU and Marcela POPESCU, Modular classes of anchored modules 79 Anatoliy PRYKARPATSKI, Ergodic and spectral properties of Lagrangian and

Hamiltonian dynamical systems and their adiabatic perturbations . . . . . . 81 Tomasz RYBICKI, Infinite dimensional Lie theory by means of the evolution map-

ping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 János SZENTHE, On the set of geodesic vectors of a left-invariant metric …. 83 Andrzej ZAJTZ, On the stability of smooth dynamical systems and diffeomorphisms 85

FOREWORD

This is the conference of the cycle initiated in 1998 with a meeting in Konopnica (http://im0.p.lodz.pl/konferencje) and is organized as in 1999 in Krynica from 29.04.2001 to 5.05.2001 (Poland). Krynica is a well known resort situated in Beskidy Mountains.

The main purpose of the conference is to present an overwiew of principal directions of research conducted in differential geometry, topology and analysis on manifolds and their applications, mainly (but not only) to Lie algebroids and related topics.

We would like to attract attention to:

• Riemannian, symplectic and Poisson manifolds,

• Lie groups, Lie groupoids, Lie algebroids and Lie-Rinehart algebras,

• foliations,

• characteristic classes.

The organizers of the conference are grateful to the following sponsors:

• Rector of the Technical University of Łódź,

• Rector of the Jagiellonian University,

• Rector of the Stanisław Staszic University of Mining and Metallurgy,

• State Committee for Scientific Research.

ORGANIZERS

3rd International Conference GEOMETRY AND TOPOLOGY OF MANIFOLDS is organized by

□ Institute of Mathematics of the Technical University of Łódź

Jan Kubarski [Chairman of the committee], kubarski@ck-sg.p.lodz.pl

□ Institute of Mathematics of the Jagiellonian University, Cracow

Robert Wolak, wolak@im.uj.edu.pl

□ Faculty of Applied Mathematics of the Stanisław Staszic University of Mining and Metallurgy, Cracow

Tomasz Rybicki, tomasz@uci.agh.edu.pl

Scientific Committee

□ Dmitri Alekseevsky (Hull, UK) □ Ivan Belko (Minsk, Belarus) □ Ronald Brown (Bangor, UK) □ Stanisław Brzychczy (Cracow, Poland) □ José F. Cariñena (Zaragoza, Spain) □ Janusz Grabowski (Warsaw, Poland) □ Július Korbaš (Bratislava, Slovakia) □ Oldřich Kowalski (Prague, Czech Republic) □ Alexandr Mishchenko (Moscow, Russia) □ Peter Nagy (Debrecen, Hungary) □ Jean Pradines (Toulouse, France) □ János Szenthe (Budapest, Hungary)

LIST OF PARTICIPANTS

1. ABIB, Renee, UMR CNRS 6085, Laboratoire Raphael SALEM, Mathématiques - Site Colbert, Université de ROUEN, 76821 Mont Saint Aignam Cedex, FRANCE, e-mail: Renee.Abib@univ-rouen.fr

2. ALEKSEEVSKY, Dimitri, Hull, UNITED KINGDOM e-mail: D.V.Alekseevsky@maths.hull.ac.uk

3. BALCERZAK, Bogdan, Institute of Mathematics, Technical University of Łódź, aleja Politechniki 11, 90-924 Łódź, POLAND, e-mail: bogdan@ck-sg.p.lodz.pl

4. BELKO, Ivan, Belarusian State Economics University, High Mathematics Department, 26 Partizansky avenue, 220070 Minsk, BELARUS, e-mail: hmd@bseu.by, niipulm@ bcsmi.minsk.by

5. BENAMEUR, Moulay–Tahar, Pennsylwania State University and Institut Desargues, Lyon, FRANCE, e-mail: benameur@desargues.univ–lyon1.fr

6. BERESTOVSKII, Valerii, Mathematical Department of Omsk State University, Laboratory of Geometry and Topology, RUSSIA, e-mail: berest@univer.omsk.su

7. BOBIEŃSKI, Marcin, Department of Mathematical Methods in Physics, University of Warsaw, ul. Hoża 74, 00-682 Warszawa, POLAND, e-mail: Marcin.Bobienski@ fuw.edu.pl

8. BOROWIEC, Andrzej, Institute of Theoretical Physics, Wrocław University, pl. M. Borna 9, 50-204 Wrocław, POLAND, e-mail: borow@ift.uni.wroc.pl

9. BROWN, Ronald, School of Informatics, Mathematics Division, University of Wales, Bangor, Dean St., Bangor, Gwynedd LL57 1UT, UNITED KINGDOM,

e-mail: r.brown@bangor.ac.uk, http://www.bangor.ac.uk/~mas010/

10. BRZYCHCZY, Stanisław, Faculty of Applied Mathematics of the University Mining and Metallurgy (AGH), Cracow, al. A. Mickiewicza 30, 30-059 Kraków, POLAND, e-mail: brzych@uci.agh.edu.pl

11. CAMACARO PEREZ, Jaime, Departamento de Física Teórica, Facultad de Ciencias, Universidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, SPAIN, e-mail: jcama@wigner.unizar.es

12. CARIÑENA, José F., Departamento de Física Teórica, Facultad de Ciencias, Universidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, SPAIN, e-mail: jfc@posta.unizar.es

13. DESZCZ, Ryszard, Department of Mathematics, Agricultural University of Wrocław, ul. Grunwaldzka 53, 50-357 Wrocław, POLAND, e-mail: rysz@ozi.ar.wroc.pl

14. DOMITRZ, Wojciech, Faculty of Mathematics and Information Science, Warsaw University of Techology, plac Politechniki 1, 00-661 Warszawa, POLAND, e-mail: domitrz@alpha.mini.pw.edu.pl

15. EWERT-KRZEMIENIEWSKI, Stanisław, Technical University of Szczecin, al. Piastów 17, 70-310 Szczecin, POLAND, e-mail: ewert@arcadia.tuniv.szczecin.pl 16. FEDORCHUK, Vasyl, Institute of Mathematics, Pedagogical Academy,

ul. Podchora̧żych 2, PL – 30-084 Kraków, Pidstryhach Institute of Applied Problems of Mechanics and Mathematics, National Ukrainian Academy of Sciences, Naukowa 3b, 79-053, L’viv, UKRAINE, e-mail: vas fedorchuk@yahoo.com

17. FERNANDES, Rui Loja, Departamento de Matematica, Instituto Superior Tecnico, 1049-001 Lisbon, PORTUGAL, e-mail: rfern@math.ist.utl.pt,

http://www.math.ist.utl.pt/~rfern

18. GENEROWICZ, Małgorzata, Institute of Mathematics, Technical University of Łódź, aleja Politechniki 11, 90-924 Łódź, POLAND, e-mail: mgenerow@ck-sg.p.lodz.pl

19. GRABOWSKI, Janusz, Mathematical Institute, Polish Academy of Sciences, ul. Śniadeckich 8, P.O. Box 137, 00-950 Warszawa, POLAND, e-mail: jagrab@mimuw.edu.pl

20. HAJDUK, Bogusław, Mathematical Institute, University of Wrocław, plac Grunwaldzki 2/4, 50-384 Wrocław, POLAND, e-mail: hajduk@math.uni.wroc.pl

21. HALBOUT, Gilles, Institut de Recherche Mathématique Avancée, Université Louis Pasteur, 7, rue René Descartes, 67084 Strasbourg Cedex, FRANCE, e-mail: halbout@irma.u.strasbg.fr

22. HALL, Graham, University of Aberdeen, Department of Mathematical Sciences, Meston Building, Aberdeen AB24 3UE, SCOTLAND, UNITED KINGDOM, e-mail: g.hall@maths.abdn.ac.uk

23. HAUSMANN, Jean-Claude, University of Geneva, SWITZERLAND, e-mail: hausmann@math.unige.ch

24. ITSKOV, Vladimir, Department of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church St., Minneapolis, MN 55455, USA, e-mail: itskov@math.umn.edu

25. KADOBIANSKI, Roman, National Technical University of Ukraine ”Kiev Polytechnic Institute”, UKRAINE, e-mail: tskafits@adam.kiev.ua

26. KOLÁŘ, Ivan, Department of Algebra and Geometry, Masaryk University, Janačkovo nam. 2a, CZ 66295, Brno, CZECH REPUBLIC, e-mail: kolar@math.muni.cz

27. KONDERAK, Jerzy, Dipartimento di Matematica, Università di Bari, ITALY, e-mail: konderak@pascal.dm.uniba.it

28. KORBAŠ, Július, Department of Algebra, Faculty of Mathematics and Physics, Comenius University, Mlynská dolina, SK–842 15, Bratislava, SLOVAKIA, e-mail: korbas@fmph.uniba.sk

29. KOWALSKI, Oldřich, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, CZECH REPUBLIC, e-mail: kowalski@karlin.mff.cuni.cz,

http://www.karlin.mff.cuni.cz/~kowalski/

30. KUBARSKI, Jan, Institute of Mathematics, Technical University of Łódź, al. Politechniki 11, 90-924 Łódź, POLAND, e-mail: kubarski@ck-sg.p.lodz.pl

31. KUREK, Jan, Institute of Mathematics, Maria Curie–Skłodowska University, pl. M. Curie–Skłodowskiej 1, 20-031 Lublin, POLAND, e-mail: kurek@golem.umcs.lublin.pl

32. KURES, Miroslav, Brno University of Technology, Technicka 2, 61669 Brno, CZECH REPUBLIC, e-mail: kures@mat.fme.vutbr.cz 33. KUSHNIREVITCH, Vitaly, National Technical University of Ukraine ”Kiev Polytechnic

Institute”, UKRAINE, e-mail: tskafits@adam.kiev.ua

34. KWAŚNIEWSKI, Andrzej Krzysztof, Institute of Computer Science, University of Białystok, ul. Sosnowa 64, 15-887 Białystok, POLAND, e-mail: kwandr@uwb.edu.pl,

http://wwwzft.uwb.edu.pl/kwandr/AKK.html

35. MIKULSKI, Włodzimierz, Institute of Mathematics, Jagiellonian University, ul. W. Reymonta 4, 30-059 Kraków, POLAND

36. MISHCHENKO, Alexandr, Moscow State University, Department of Mathematics, Vorobjovy Gory, 117234, Moscow, RUSSIA, e-mail: asmish@mech.math.msu.su

37. MORMUL, Piotr, Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warszawa, POLAND, e-mail: mormul@mimuw.edu.pl

38. MOZGAWA, Witold, Institute of Mathematics, Maria Curie–Skłodowska University (UMCS), plac M. Curie–Skłodowskiej 1, 20-031 Lublin, POLAND, e-mail:mozgawa@ golem.umcs.lublin.pl

39. NAGY, Peter Tibor, Debrecen University, Institute of Mathematics, Department of Geometry, H-4010 DEBRECEN, P.O.B. 12, HUNGARY, e-mail: nagypeti@math.klte.hu

40. NEGREIROS, Caio, University of Campinas, Rua Uruguaiana, 405 Ap.42, Campinas, SP, BRASIL, e-mail: caione@ime.unicamp.br

41. NGUIFFO BOYOM, Michel, Departement de Mathématiques, Université Montpellier2, FRANCE, e-mail: boyom@math.univ-montp2.fr

42. OLSZAK, Zbigniew, Institute of Mathematics, Technical University of Wrocław, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, POLAND, e-mail: olszak@im.pwr.wroc.pl

43. OPOZDA, Barbara, Institute of Mathematics, Jagiellonian University, Cracow, ul. W. Reymonta 4, 30-059 Kraków POLAND, e-mail: opozda@im.uj.pl

44. PIA̧TKOWSKI, Andrzej, Institute of Mathematics, Technical University of Łódź, aleja Politechniki 11, 90-924 Łódź, POLAND, e-mail: andpiat@ck-sg.p.lodz.pl

45. POPESCU, Paul, Department of algebra and geometry, University of Craiova, 13, ’Al.I.Cuza’ st., Craiova, 1100, ROMANIA, e-mail: paulpopescu@email.com

46. PRYKARPATSKI, Anatoliy K., Deptartment of Applied Mathematics at the AGH, 30-059 Kraków, POLAND, and Dept. for Nonlinear Mathematical Analysis at the Academy of Sciences of Ukraine, L’viv 79052, UKRAINE, e-mail: prykanat@cybergal.com

47. RYBICKI, Tomasz, Faculty of Applied Mathematics of the University of Mining and Metallurgy (AGH), Cracow, al. A. Mickiewicza 30, 30-059 Kraków, POLAND, e-mail: tomasz@uci.agh.edu.pl

48. SLUKA, Karina, Institute of Mathematics, Technical University of Wrocław, Wybrzeże Wyspianskiego 27, 50-370 Wrocław, POLAND, e-mail: sluka@im.pwr.wroc.pl

49. STASICA, Anna, Laboratoire de Mathématiques, Université de Savoie and Institute of Mathematics of the Jagiellonian University, ul. Reymonta 4, 30–059 Kraków, POLAND, e-mail: astasica@im.uj.edu.pl 50. SZENTHE, János, Dept. of Geometry, Eötvös Univ. Kecskeméti u. 10-12, Budapest,

H-1053, HUNGARY, e-mail: szenthe@ludens.elte.hu

51. TOMÁŠ, Jiří, Department of Mathematics, Technical University Brno, Žižkova 17, 602 00 Brno, CZECH REPUBLIC, e-mail: Tomas.J@fce.vutbr.cz

52. URBAŃSKI, Paweł, Division of Mathematical Methods in Physics, University of Warsaw, ul. Hoża 74, 00-682 Warszawa, POLAND, e-mail: urbanski@fuw.edu.pl

53. WALAS, Witold, Institute of Mathematics, Technical University of Łódź, aleja Politechniki 11, 90-924 Łódź, POLAND, e-mail: walwit@ck-sg.p.lodz.pl

54. WOLAK, Robert, Institute of Mathematics, Jagiellonian University, ul. Reymonta 4, 30-059 Kraków, POLAND, e-mail: wolak@im.uj.edu.pl

55. ZAJTZ, Andrzej, Institute of Mathematics, Pedagogical Academy of Cracow, ul. Podchora̧żych 2, 30-084 Kraków, POLAND, e-mail: smzajtz@cyf-kr.edu.pl

TITLES OF LECTURES

1. BELKO, Ivan  The Structure Invariants of a Transverse G-Structure

2. BENAMEUR, Moulay–Tahar  Homology of Complete Symbols and

Non-commutative Geometry

3. BOROWIEC, Andrzej  Metric-polynomial structures from gravitational

Lagrangians

4. CAMACARO, Jaime  Application of Lie algebroids on field theories

5. DESZCZ, Ryszard  On some class of semisymmetric manifolds

6. EWERT-KRZEMIENIEWSKI,
Stanisław  On a class of Ricci-recurrent manifolds

7. FEDORCHUK, Vasyl  On invariants of continuous subgroups of

the generalized Poincaré group P (1, 4)

8. FERNANDES, Rui Loja  1. Integrating Lie algebroids to Lie groupoids

2. Lie algebroids and characteristic classess

9. GRABOWSKI, Janusz  Jacobi structures revisited

10. HALL, Graham  Orbits of Symmetries in General Relativity

11. KADOBIANSKI, Roman
and KUSHNIREVITCH, Vitaly  Differential Geometry Structure for Formal Maps

12. KOLÁŘ, Ivan  Functorial prolongations of principal bundles

and Lie algebroids

13. KOWALSKI, Oldřich  Generalized Symmetric Spaces

14. KUBARSKI, Jan  Cohomology of flat connections in some Lie algebroids

15. KUSHNIREVITCH, Vitaly
and KADOBIANSKI, Roman  Configuration Spaces and Algebroids

TITLES OF LECTURES

16. KWAŚNIEWSKI, Andrzej K.  Extended finite operator calculus – an example of

algebraization of analysis

17. MASLOV V. P.
and MISHCHENKO, Alexandr  Geometry of Lagrangian manifolds in Thermodynamics

18. MISHCHENKO, Alexandr  The Hirzebruch formula with nonflat coefficients

19. MORMUL, Piotr  Geometry of Goursat flags and their singularities of

codimension 2

20. NAGY, Peter Tibor  2-divisible Bruck loops and exponential affine

symmetric spaces

21. NEGREIROS, Caio  On (1,2)-symplectic structures on flag manifolds and

loop groups

22. NGUIFFO BOYOM, Michel  KV-cohomology of contact manifolds

23. PIA̧TKOWSKI, Andrzej  On prefoliations of the K-differential space

24. POPESCU, Paul  Submodules of vector fields and algebroids

25. POPESCU, Paul
and POPESCU, Marcela  Modular classes of anchored modules

26. PRYKARPATSKI, Anatoliy  Ergodic and spectral properties of Lagrangian and

Hamiltonian dynamical systems and their adiabatic perturbations

27. RYBICKI, Tomasz  Infinite dimensional Lie theory by means of

the evolution mapping

28. SZENTHE, János  On the set of geodesic vectors of a left-invariant metric

29. ZAJTZ, Andrzej  On the stability of smooth dynamical systems and

diffeomorphisms

ABSTRACTS. PAPERS

THE STRUCTURE INVARIANTS OF A TRANSVERSE G-STRUCTURE

IVAN BELKO

Abstract

The problems of equivalence and integrability of G-structure were considered by many authors. Ch.Ehresmann, E.Cartan, S.Chern, V.Guillemin, D.Bernard and many others studied this problems. The solution of the equivalence problem is based on construction of structure invariants and cohomology classes correspondent. It is possible to pick out two main approaches to construction of the structure tensor, wich are different in their methods. We can call one of them geometrical, it uses the fundamental form and the forsion of the connection on principal bundle of the frames. Another approach is caracterised by use of differential operators and their Spencer cohomologies. The Ngo van Que paper [1] is an example of this.

Our goal is the construction of a structure invariants for a kth order foliated transversal structure on foliated manifold. Geometrical construction of such tensor for the first- order transversal structure is given in R.Wolak paper [2].

The particularity of our approach is in using of foliated Lie algebgoid anddevelopment of Spencer cohomology method to transversal differential operators. For a foliated manifold (B, F) the transversal k-jets of local diffeomorfisms forms the foliated Lie groupoid Π^{k}(B). His Lie algebroid can be identified with the foliated Lie algebroid Ĵ^{k}(T B) of transversal k-jets of foliated vector filds. In this Lie algebroid the partial plate connection τ : T F → Ĵ^{k}(T B) is defined by canonical way. The natural projection by Ker τ determines the exact sequance of foliated vector bundles

0 → Ĵ_{o}^{k}(T B) → Ĵ^{k}(T rB) → T rB → 0.

The adjoint Lie algebra bundle Ĵ_{o}^{k}(T B) = Ĵ_{o}^{k}(T rB) plays an important role in construction of a structure invariants.

A regular infinitesimal transversal kth order structure on a foliated manifold is a regular section of the bundle associated to Lie groupoid Π^{k}(B). Such a bundle is foliated and permits to distinguish the foliated sections and structures.

A regular structure S defines a Lie subgroupoid Π^{k}(S) in Π^{k}(B). His Lie subalgebroid is a vector bundle and can be considered as a system of linear differential equations

E^{k}(S) ⊂ Ĵ^{k}(T B).

On the base of the exacte sequance of vector bundles is built the transversal Spencer operator

D̂ : Γ(Ĵ^{k}T B) → Γ(Ĵ^{k−1}T B ⊗ T r^{∗}B).

This operator defines a transversal cohomologycal sequance

δ : T rB ⊗ S^{k+1}(T r^{∗}B) ⊗ Λ^{p}(T r^{∗}B) → T rB ⊗ S^{k}(T r^{∗}B) ⊗ Λ^{p+1}(T r^{∗}B).

In the terms of these cohomologies is defined the type and the degree of a structure S. For a connection in Lie algebroid Ĵ^{k}(T B) also in its Lie subalgebroid E^{k}(S) can be defined the torsion. The class of transversal δ-cohmology correspondent is an obstacle to the integrability of a foliated first-order G-structure S.

REFERENCES

[1 ] Ngo van QUE , Du prolongement des espaces fibrés et des structures infinitésimales, Ann. Inst. Fourier (Grenoble), 17, 1 (1967), 157-223.

[2 ] R.A.Wolak, The structure tensor of a transverse G-structure on a foliated manifold, Bollettino U.M.I. (7) 4-A (1990), 1-15.

Belarusian State Economics University High Mathem. Department 26 Partizansky avenue, 220070 Minsk BELARUS e-mail: hmd@bseu.by, niipulm@bcsmi.minsk.by

HOMOLOGY OF COMPLETE SYMBOLS AND NON-COMMUTATIVE GEOMETRY

MOULAY–TAHAR BENAMEUR and VICTOR NISTOR

Abstract

We identify the periodic cyclic homology of the algebra of complete symbols on a differential grupoid G in terms of the cohomology of S^{∗} (G), the cosphere bundle of A (G), where A (G) is the Lie algebroid of G. We also relate the Hochschild homology of this algebra with the homogenous Poisson homology of the space A^{∗} (G) \ 0 ≅ S^{∗} (G) × (0, ∞), the dual of A (G) with the zero section removed. We use then these results to compute the Hochschild and cyclic homologies of the algebras of complete symbols associated with manifolds with corners, when the corresponding Lie algebroid is rationally isomorphic to the tangent bundle.

MOULAY–TAHAR BENAMEUR Pennsylvania State University and Institut Desargues FRANCE e-mail: benameur@desargues.univ–lyon1.fr

VICTOR NISTOR Institut Desargues and Pennsylvania State University University Park, PA 16802 e-mail: nistor@math.psu.edu

http://www.math.psu.edu/nistor/

METRIC-POLYNOMIAL STRUCTURES FROM GRAVITATIONAL LAGRANGIANS

ANDRZEJ BOROWIEC

Abstract

We study these metric-polynomial structures on manifold which arises as extremals of the Palatini variational principle for some class of gravitational Lagrangians. They can be described in the following way.

Let (M, g, Γ) be a n-dimensional (pseudo-) Riemannian manifold (M, g) equipped with a symmetric (i.e. torsion-free) connection Γ. Define a (1, 1) tensor field concomitant

S_{ν}^{μ} ≡ S_{ν}^{μ}(g, Γ) ≐ g^{μα}R_{(αν)}(Γ)

where R_{(αν)}(Γ) denotes the symmetric part of the Ricci tensor of Γ. Consider a family of scalar-valued concomitant

s_{k} ≡ s_{k}(g, Γ) ≐ tr S^{k}

the so-called Ricci scalar of order k, k = 1, ⋯, n. For F being an arbitrary (differentiable) real-valued function of n-variables one can define the corresponding Lagrangian of the Ricci type

L_{F} ≐ \sqrt{g} F (s_{1}, ⋯, s_{n})

Applying now the Palatini variational principle we arrive to the following results:

• the connection Γ is a Levi-Civita connection for some pseudo-Riemannian Einstein metric h,

• the (1, 1) tensor field S satisfies a polynomial equation

w_{F} (S) = 0

for some polynomial function w_{F} (t) of constant coefficients,

• the metric h and polynomial structure S are compatible in a sense

h(SX, Y) = h(X, SY)

for each pair of tangent vector fields (X, Y) on M.

In particular, for f being a function of one variable the Lagrangian L_{f} = \sqrt{g} f(s_{1}) reconstructs the Einstein theory. For L_{f} = \sqrt{g} f(s_{2}), besides the Einstein equation, one gets a pseudo-Riemannian almost product structure (S^{2} = I) and/or an almost-complex anti-Hermitian structure (S^{2} = −I). Some other examples will be also considered.

Institute of Theoretical Physics, Wrocław University

pl. M. Borna 9, 50-204 WROCŁAW, Poland

e-mail: borow@ift.uni.wroc.pl

APPLICATION OF LIE ALGEBROIDS ON FIELD THEORIES

JAIME CAMACARO

Abstract

We will show some examples of how the Lie algebroids can play an interesting role in the study of gauge fields theories. Our talk will be based on four examples: Yang–Mills theory, Topological sigma models, Poisson sigma models and open bosonic string, where the Lie algebroid structure can be easily recognized and used to obtain an algebra which take in consideration the gauge structure.

Jaime CAMACARO Departamento de Física Teórica, Facultad de Ciencias Universidad de Zaragoza SPAIN e-mail: jcama@wigner.unizar.es

Thanks: JRC acknowledges the financial support of the Agencia Española de Cooperación Internacional, under an AECI scholarship

Keywords: BV master equation, Gauge theories, Lie algebroids

ON A CLASS OF RICCI-RECURRENT MANIFOLDS

STANISŁAW EWERT-KRZEMIENIEWSKI

1 Introduction

Following Prvanivić ([P]), a semi-Riemannian manifold (M, g) will be called conformally quasi-recurrent if its Weyl conformal curvature tensor C satisfies

\begin{matrix}∇_{Z}C(X, Y, V, W) = w(Z)C(X, Y, V, W)+ \\ p(X)C(Z, Y, V, W) + p(Y)C(X, Z, V, W)+ \\ p(V)C(X, Y, Z, W) + p(W)C(X, Y, V, Z) \end{matrix}  (1.1.1)

for some 1-forms w, p. In condition considered originally by Prvanović w = 2p. However, the last relation together with (1.1.1) implies that for the tensor C ( and in fact for any generalized curvature tensor satisfying (1.1.1)) the second Bianchi identity must hold. The aim of this note is to give a classification of conformally quasi-recurrent manifolds in the sense of (1.1.1) which are simultaneously Ricci-recurrent, i. e. those the Ricci tensor S satisfies

∇S = b ⊗ S

for some 1-form b.

For a generalized curvature tensor B define B̃ by

g (B̃ (X, Y) V, W ) = B (X, Y, V, W) .

Then for a (0, k) tensor field T, k ≥ 1, and (0, 2) tensor field S we define the tensor fields B · T and Q(S, T) by the formulas

\begin{matrix}(B · T) (X_{1}, ...X_{k}; X, Y) = \\ −T (B̃ (X, Y) X_{1}, X_{2}, ..., X_{k}) − ... − T (X_{1}, ..., X_{k−1}, B̃ (X, Y) X_{k}) , \end{matrix}

\begin{matrix}Q (S, T) (X_{1}, ...X_{k}; X, Y) = \\ T ((X ∧_{S} Y) X_{1}, X_{2}, ..., X_{k}) + ... + T (X_{1}, ..., X_{k−1}, (X ∧_{S} Y) X_{k}) , \end{matrix}

where

(X ∧_{S} Y) Z = S(Y, Z)X − S(X, Z)Y.

If the tensors R · R and Q(S, R) are linearly dependent then the manifold is said to be Ricci-generalized pseudosymmetric one ([D-D]). It is obvious that any semisymmetric as well as any Ricci flat manifold is Ricci generalized pseudosymmetric. The manifold (M, g) is Ricci-generalized pseudosymmetric iff the relation

R · R = LQ(S, R)  (1.1.2)

holds on the set {x ∈ M, Q(S, R) (x) ≠ 0} , L being a function on M. Remark that the relation (1.1.2) with L = 1 is of particular importance.

All manifolds under consideration are assumed to be smooth Hausdorff connected and their metrics are not assumed to be definite.

2 Results

The first Lemma shows the difference between the 1-forms w and p :

Lemma 2.1. Suppose that at a point of the manifold M relation (1.1.1) holds. Then

p_{r}C_{ijk}^{r} = 0,
w_{r}C_{ijk}^{r} = C_{ijk,r}^{r},

\begin{matrix}C_{hijk,[lm]} = Δw_{lm}C_{hijk} + p_{hm}C_{lijk} + p_{im}C_{hljk} + p_{jm}C_{hilk} + p_{km}C_{hijl}− \\ p_{hl}C_{mijk} − p_{il}C_{hmjk} − p_{jl}C_{himk} − p_{kl}C_{hijm}, \end{matrix}

where Δw_{lm} = w_{l,m}−w_{m,l}, p_{hm} = p_{h,m}−p_{h}p_{m} and comma denotes covariant differentiation.

From the so called Patterson identity we have

Proposition 2.2. Let M be a 4-dimensional manifold with nowhere vanishing Weyl conformal curvature tensor C. If C satisfies (1.1.1), then M is conformally recurrent manifold, precisely

∇C = (w + 2p) ⊗ C.

In the sequel we shall assume the following hypothesis:

(H) M is a Ricci − recurrent manifold with nowhere vanishing W eyl conformal curvature tensor and Ricci tensor. Moreover, the W eyl conformal curvature tensor satisfies (1.1.1), p does not vanishes on a dense subset and the Ricci tensor is not parallel.

By hypothesis, M admits a covector field b satisfying

S_{ij,k} = b_{k}S_{ij}, S_{ij,kl} = (b_{k,l} + b_{k}b_{l})S_{ij}, S_{ij,[kl]} = Δb_{kl}S_{ij},

where Δb_{kl} = b_{k,l} − b_{l,k}.

Proposition 2.3. Let M (dimM > 3) be a Ricci-recurrent manifold with non-parallel Ricci tensor and suppose that M admits a covector field p with properties:

i) p does not vanish on a dense subset of M;

ii) p_{r}C_{ijk}^{r} = 0 on M.

Then the scalar curvature of M vanishes.

Lemma 2.4. Under hypothesis (H) relations

p_{r}S_{h}^{r} = 0,
db = 0
S_{mr}C_{ijk}^{r} = 0

hold on M.

From the above Lemma it follows Corollary 2.5. Under hypothesis (H)

d(w + 2p) = 0

holds on M.

Making use of the Corollary 2.5 one can deduce

Proposition 2.6. Assume that on a manifold (M, g) hypothesis (H) is satisfied.

If t = w − 2p = 0 on (M, g), then the manifold is conformally related to a nonconformally flat conformal symmetric one (M, Exp (2f) g). Conversely, if (M, g) is conformally related to a non-conformally flat conformally symmetric one, then w − 2p = 0 .

Proposition 2.7. Suppose that on a manifold M, (dimM > 4), hypothesis (H) is satisfied. If b_{l} ≠ 0 at a point x ∈ M, then on some neighbourhood of x there exists null (i.e. isotropic) parallel vector field

v_{i} = Exp[\frac{−1}{2}b] k_{i}, ∂_{i}b = b_{i},

related to the Ricci tensor by

S_{ij} = ϵ k_{i}k_{j}, |ϵ| = 1.

Define the tensor

\begin{matrix}Q(S, C)_{hiqtpj} = S_{hp}C_{jiqt} − S_{hj}C_{piqt} + S_{ip}C_{hjqt} − S_{ij}C_{hpqt}+ \\ S_{qp}C_{hijt} − S_{qj}C_{hipt} + S_{tp}C_{hiqj} − S_{tj}C_{hiqp}. \end{matrix}

It is well known, that if the scalar curvature vanishes and the rank of the Ricci tensor is one, then

Q(S, C)_{hiqtpj} = Q(S, R)_{hiqtpj}

Lemma 2.8. Suppose that on a manifold M, (dimM > 4), hypothesis (H) is satisfied. Then

t_{r}b^{r} · Q(S, C)_{hiqtpj} = 0.

Lemma 2.9. Suppose that on a manifold M, (dimM > 4), hypothesis (H) is satisfied and

t_{r}b^{r} = 0.

Then

(n − 4) b_{r}b^{r} = 0,

t_{l}C_{hijk} + t_{j}C_{hikl} + t_{k}C_{hilj} = 0

and

t_{l} (C_{tqlr}C_{jih}^{r} − C_{tqjr}C_{lih}^{r}) = 0

Proposition 2.10. Suppose that on a manifold M, (dimM ≥ 4), hypothesis (H) is satisfied. If at x ∈ M, t_{r}b^{r} (x) ≠ 0 (hence b_{l}(x) ≠ 0), then on some neighbourhood of x the Riemann-Christoffel curvature tensor has the form

R_{hijk} = k_{i}k_{j}S_{hk} − k_{i}k_{k}S_{hj} + k_{h}k_{k}S_{ij} − k_{h}k_{j}S_{ik},

where S_{ij} = z^{r}z^{s}R_{rijs}, z^{r}k_{r} = 1.

As a consequence of Propositions 2.3, 2.7 and 2.10 we obtain

Corollary 2.11. Suppose that on a manifold M, (dimM > 4), hypothesis (H) and and Q (S, C)_{hiqtpj} = 0 hold. If b_{l} ≠ 0 at a point x ∈ M, then on some neighbourhood of x the metric g is of the Walker type. Moreover, M is semi-symmetric, i.e. R · R = 0.

On the other hand, if Q (S, C)_{hiqtpj} ≠ 0, then Lemma 2.9 results in

Corollary 2.12. Let on a manifold M, (dimM > 4), hypothesis (H) be satisfied. Suppose moreover that Q (S, C)_{hiqtpj} ≠ 0 and w − 2p ≠ 0. Then

(R · R)_{hiqtpj} = Q(S, R)_{hiqtpj}.

Thus we conclude with the following

Theorem 2.13. A conformally quasi-recurrent and Ricci-recurrent manifold of dimension n > 4 with nowhere vanishing Weyl conformal curvature tensor and Ricci tensor nonconformally related to a conformally symmetric one must be necessary Ricci-generalized pseudosymmetric manifold.

References

[D-D] F. Defever, R. Deszcz, On semi-Riemannian manifolds satisfying the condition R·R = Q(S, R), Geometry and Topology of Submanifolds, 3(1991), 108-130, World Sci., Singapore.

[P] M. Prvanović, Conformally quasi-recurrent manifolds, Finsler and Lagrange spaces, Proc. 5th Natl. Semin., Hon. 60th Birthday R. Miron, Braşov/Rom. 1988, (1988), 321-328.

Institute of Mathematics

Technical University of Szczecin

al. Piastów 17, 70-310 Szczecin

POLAND

e-mail: ewert@arcadia.tuniv.szczecin.pl

ON INVARIANTS OF CONTINUOUS SUBGROUPS OF THE GENERALIZED POINCARÉ GROUP P (1, 4)

VASYL FEDORCHUK

Abstract

For all continuous subgroups of the group P (1, 4) the invariants in the five-dimensional Minkowski space M(1, 4) have been constructed. The invariants obtained are one-, two-, three and four-dimensional.

On the base of the invariants obtained the nonsingular manifolds in the spaces M(1, 3)× R and M(1, 4) × R invariant under nonconjugate subgroups of the group P (1, 4) have been described.

The manifolds obtained have already been used for the symmetry reduction of some important equations of the theoretical physics in the spaces M(1, 3) × R and M(1, 4) × R.

Institute of Mathematics, Pedagogical Academy ul. Podchora̧żych 2, PL – 30-084 Kraków Pidstryhach Institute of Applied Problems of Mechanics and Mathematics National Ukrainian Academy of Sciences Naukowa 3b, 79-053, L’viv UKRAINE e-mail: vas fedorchuk@yahoo.com

INTEGRATING LIE ALGEBROIDS TO LIE GRUPOIDS

RUI LOJA FERNANDES

Abstract

I will report on some recent joint work with Marius Crainic (Utrecht), where we give the obstructions for integrating Lie algebroids to Lie grupoids [2]. This puts into a new perspective the work of Cattaneo and Felder [1] for the special case of Poisson manifolds and the ”new” proof of Lie’s third theorem given by Duistermaat and Kolk in [3].

References

[1] A. Cattaneo and G. Felder, Poisson sigma models and symplectic groupoids, math.DG/0003023

[2] M. Crainic and R. L. Fernandes, Integrating Lie algebroids to Lie groupoids, in preparation (it will be available in xxx by the time of the meeting)

[3] J. Duistermaat and J. Kolk, Lie groups, Springer Universitytext, 1999.

Rui Loja FERNANDES

Departamento de Matematica Instituto Superior Tecnico 1049-001 Lisbon PORTUGAL e-mail: rfern@math.ist.utl.pt

http://www.math.ist.utl.pt/~rfern

JACOBI STRUCTURES REVISITED

JANUSZ GRABOWSKI

Abstract

A lifting procedure of first-order multi-differential operators is defined which maps the Nijenhuis-Richardson into the Schouten bracket. This is a way of associating canonically a Lie algebroid with any local Lie algebra structure on a 1–dimensional vector bundle.

Mathematical Institute, Polish Academy of Sciences ul. Śniadeckich 8 P.O. Box 137, 00-950 Warszawa POLAND e-mail: jagrab@mimuw.edu.pl

DIFFERENTIAL GEOMETRY STRUCTURE FOR FORMAL MAPS

ROMAN KADOBIANSKI and VITALY KUSHNIREVITCH

Abstract

Usually, differential geometry is formulated in the terms of smooth maps of smooth manifolds. The conception proposed by I.M.Gel’fand and Yu.L.Daletskii is to replace smooth manifolds by formal one. The formal manifold is the pair (𝔄, M), where 𝔄 is Lie algebra and

M is a module over it. Let L_{k}(X, Y) be k-linear map from X to Y; (X, Y) = ∏_{k=1}^{∞} L_{k}(X, Y).

Formal map is defined as sequence a = (a_{1}, a_{2}, …) ∈ (X, Y), a_{k} ∈ L_{k}(X, Y). The natural further step is to use formal maps instead of smooth ones. It turns out that Lie algebra structure and module over it can be defined in the space of formal maps. So, differential geometry of formal maps can be constructed. For example, such construction is useful in nonlinear partial diferential equations theory and for different nontrivial dynamical systems in physics.

Roman KADOBIANSKI National Technical University of Ukraine ”Kiev Polytechnic Institute” UKRAINE e-mail: tskafits@adam.kiev.ua

Vitaly KUSHNIREVITCH National Technical University of Ukraine ”Kiev Polytechnic Institute” UKRAINE e-mail: tskafits@adam.kiev.ua

GENERALIZED SYMMETRIC SPACES

OLDŘICH KOWALSKI

Abstract

1 Introduction

Let (M, g) be a Riemannian symmetric space. Then for any x ∈ M, there exists an isometry s_{x} : M ⟶ M such that x is an isolated fixed point of s_{x} and s_{x}^{2} = Id. Then we have

(s_{x})_{∗x} = (−Id)_{x}
v_{x} ↦ −v_{x}

Now we consider a generalization of the notion of Riemannian symmetric spaces: Let (M, g) be a Riemannian manifold. We consider only the following condition.

∀x ∈ M, ∃s_{x} : M ⟶ M : isometry s.t. x : isolated fixed point

Such an isometry s_{x} is called a generalized symmetry. And a set of generalized symmetries {s_{x} | x ∈ M} is called a (Riemannian) s-structure.

Lemma 1.1. If {s_{x} | x ∈ M} on (M, g) is an s-structure, then the closure Cl({s_{x}}) of the group generated by all s_{x}, x ∈ M, in the Lie group I(M, g) (the full isometry group) acts transitively on (M, g).

Theorem 1.1 (Brickel). Each (M, g) admitting an s-structure is homogeneous and thus real analytic.

Here, “homogeneous” means that ∀x, y ∈ M, ∃φ : M ⟶ M : isometry, such that φ(x) = y.

The proof is elementary and interesting, but not short.

Affine case analogue:

Theorem 1.2 (Ledger). Let (M, ∇) be an affine manifold (with an affine connection ∇). Let there exists a family {s_{x} | x ∈ M} of generalized affine symmetries. If the map x ↦ s_{x} is smooth, then (M, ∇) is a homogeneous affine manifold.

In general, it is an open problem whether (M, ∇) is still homogeneous if the map x ↦ s_{x} is not smooth. (Elementary but interesting question).

Regularity condition:

s_{x} ∘ s_{y} = s_{z} ∘ s_{x} z = s_{x}(y) ∀x, y ∈ (M, g)

This condition is satisfied for each symmetric space and its standard symmetries s_{x}!

Example. In the case ℝ^{2}, we can illustrate the above condition by a picture. (In general, it is not easy to check).

[figure: x; ū = s_{x}(u); u; z = s_{x}(y); y; u′′ = s_{x}(u′); u′ = s_{y}(u)]

Definition 1. Tangent symmetry field S of {s_{x} | x ∈ M}

S_{x} := (s_{x})_{∗x} ∀x ∈ (M, g)

Lemma 1.2. The regularity condition for {s_{x} | x ∈ M} is satisfied if and only if S is invariant with respect to each s_{x}:

(s_{x})_{∗y} ∘ S_{y} = S_{s_{x}(y)} ∘ (s_{x})_{∗y} ∀x, y ∈ (M, g)

In the case of symmetric spaces,

S_{y} = (−Id)_{y}, S_{s_{x}(y)} = (−Id)_{s_{x}(y)},

and the regularity follows.

An s-structure {s_{x} | x ∈ M} which satisfies the regularity condition is called a regular s-structure. (It is a very strong condition).

Example of regular s-structures: M = ℝ^{2}, s_{x} = rotation arround x with constant angle α. It is obvious from Lemma 2 that this s-structure is regular because each tangent symmetry S_{x} is also a rotation. (This direct proof of regularity is a good excercise for high school students).

[figure: x; y]

Proposition 1.1. If {s_{x} | x ∈ M} is a regular s-structure on (M, g), then the tangent field is analytic and the map (x, y) ↦ s_{x}(y) is also analytic.

Proposition 1.2. If (M, g) admits an s-structure (resp. regular s-structure) {s_{x} | x ∈ M}, then (M, g) admits an s-structure (resp. regular s-structure) {s′_{x} | x ∈ M} of finite order. It means that ∃k ∈ ℤ, k ≥ 2 s.t.

(s′_{x})^{k} = Id for ∀x ∈ M.

Remark: It it not true for the affine case.

If there exists a regular s-structure {s_{x} | x ∈ M} of order k on (M, g), then (M, g) is called a k-symmetric space. (Without regularity, a pointwise k-symmetric space).

Thus if there exists a regular s-structure on (M, g), then for some ∃k, (M, g) is a ksymmetric space.

Regularity condition is really an additional condition but in a nontrivial sense:

1) (R.A. Marinosci [4]) If dim(M, g) ≤ 5, then every (M, g) admitting an s-structure also admit a regular s-structure.

2) S^{9} = SU(5)/SU(4) S^{9} is a geodesic sphere in (P^{5}, Fubini-Study metric) with the induced metric, which is not the standard sphere. This S^{9} admits an s-structure of order 4 but no regular s-structure at all.

There are finitely many other examples S^{13}, S^{17}, · · ·.

In the following, we always assume regularity!

(M, g) is called a generalized symmetric space if it admits a regular s-structure. (M, g) is called of order k (k ≥ 2) if it admits a regular s-structure of order k but not

of any lower order l < k.

2 Examples and Classifications

Dimension n = 3

G = ∥ \begin{matrix}e^{−z} & 0 & x \\ 0 & e^{z} & y \\ 0 & 0 & 1 \end{matrix} ∥

(solvable)

G ≅ ℝ^{3}(x, y, z)

Special invariant metrics on G:

g = e^{2z}dx^{2} + e^{−2z}dy^{2} + λ^{2}dz^{2}

where λ > 0 is an arbitrary parameter. Symmetry of order 4 at the origin:

x′ = −y, y′ = x, z′ = −z

(See B-Spaces by Takahashi).

Thses are all generalized symmetric spaces of dimension n = 3 which are not locally symmetric!

One kind of symmetric spaces:

G: compact connected Lie group Consider the coset space (G × G)/Δ(G × G), where Δ(G × G) is the diagonal of G × G, i.e., Δ(G × G) = {(g, g) | g ∈ G}. G × G/Δ(G × G) is diffeomorphic to G via the map

(g_{1}, g_{2}) ↦ g_{1}g_{2}^{−1}.

G × G acts on G by

(g_{1}, g_{2})(y) = g_{1}yg_{2}^{−1}.

The isotropy group at the origin e ∈ G is Δ(G × G).

Now define σ : G × G ⟶ G × G by

σ(g_{1}, g_{2}) = (g_{2}, g_{2}),

which is an involutive automorphism. The fixed point set is (G × G)^{σ} = Δ(G × G). σ induces a map s : G ⟶ G defined by

s(g) = g^{−1} for ∀g ∈ G.

Take a bi-invariant Riemannian metric Φ on G. Then Φ is invariant with respect to the action of G × G on G. It is also invariant with respect to s : G ⟶ G.

Then (G, Φ) is a Riemannian symmetric space in which the symmetry with respect to e is just the map

s : g ↦ g^{−1}.

The symmetry s_{x} at general x ∈ G is given by

g ↦ xg^{−1}x.

Generalization(Ledger + Obata)

G^{k+1}/ΔG^{k+1} ≅ G^{k}

via π : G^{k+1} ⟶ G^{k}, where

π(g_{1}, ⋯, g_{k+1}) = (g_{1}g_{k+1}^{−1}, ⋯, g_{k}g_{k+1}^{−1}).

Further define σ : G^{k+1} ⟶ G^{k+1} by

σ(g_{1}, ⋯, g_{k+1}) = (g_{k+1}, g_{1}, ⋯, g_{k}).

Then σ is an automorphism of order k + 1. It induces a map s : G^{k} ⟶ G^{k} defined by

s(g_{1}, ⋯, g_{k}) = (g_{k}^{−1}, g_{1}g_{k}^{−1}, ⋯, g_{k−1}g_{k}^{−1}).

Let Φ be a bi-invariant metric on G. Then Φ generates a bi-invariant metric Φ^{k+1} on G^{k+1} such that

(G^{k+1}, Φ^{k+1}) ≅ \underbrace{(G, Φ) × ⋯ × (G, Φ)}_{k+1} .

Then Φ^{k+1} induces a G^{k+1}-invariant metric Φ^{[k]} on G^{k}. The Riemannian manifold (G^{k}, Φ^{[k]}) is a (k + 1)-symmetric space. (Here Φ^{[k]} ≠ Φ^{k}). (G^{k}, Φ^{[k]}) is not a symmetric space.

Proposition 2.1. Let

a) G be compact and simple,

b) Φ = −(Killing form). Assume that τ : G^{k+1} ⟶ I(G^{k}, Φ^{[k]}) has as image the full identity component of I(G^{k}, Φ^{[k]}). Then (G^{k}, Φ^{[k]}) is not l-symmetric for any l < k + 1.

Example. If we choose G = SO(3), Φ = −(Killing form), then the assumption of the Proposition 3 can be shown to be satisfied. Hence we get

Theorem 2.1. For each k ≥ 2, there is a generalized symmetric space (M, g) of order k, i.e., k-symmetric but not l-symmetric for ∀l < k. M = G/K with semisimple group G.

Classifications:

1) All generalized symmetric spaces of order 3 with semisimple (or reductive group) G by A. Gray [1].

2) All compact generalized symmetric spaces of order 4 by J.A. Jiménez [2].

An example of generalized symmetric Riemannian spaces of solvable type:

G = ∥ \begin{matrix}e^{u_{0}} & 0 & ⋯ & 0 & x_{0} \\ 0 & e^{u_{1}} & ⋯ & 0 & x_{1} \\ ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & ⋯ & e^{u_{n}} & x_{n} \\ 0 & 0 & ⋯ & 0 & 1 \end{matrix} ∥ ≅ ℝ^{2n+1}

u_{0}, u_{1}, ⋯, u_{n}, x_{1}, ⋯, x_{n} are variables, where

u_{0} + u_{1} + ⋯ u_{n} = 0.

We define a Riemannian metric

g = ∑_{i=0}^{n} e^{−2u_{i}}(dx_{i})^{2} + a ∑_{α,β=1}^{n} du_{α}du_{β}, a > 0.

Then

1) (G, g) is generalized symmetric of order 2n + 2 (even).

2) The group of all isometries preserving the origin is finite and isometric to

(ℤ_{2})^{n+1} × S_{n+1}

where S_{n+1} is the permutation group pf n + 1 elements. Thus, the group is of order 2^{n+1}(n + 1)!.

Remark. There are also examples of generalized symmetric spaces of solvable type and odd order. For any order k, there exists an example of solvable type. This is the main obstacle to the classification of all generalized symmetric spaces (in contrary to the ordinary symmetric spaces, where the classification is known).

3 Canonical connection

Let (M, g) be a Riemannian manifold and {s_{x} | x ∈ M} a regular s-structure on M. Then there is a unique affine conncetion ˜∇ on M such that

(i) ˜∇ is invariant under all s_{x}, (ii) ˜∇S = 0 (where S is the tangent symmetry field defined by S_{x} := (s_{x})_{∗x} x ∈ M).

In the explicit form, ˜∇ is given by the Ledger’s formula

˜∇_{X}Y = ∇_{X}Y − (∇_{(I−S)^{−1}X}S)(S^{−1}Y) for X, Y ∈ 𝔛(M)

where ∇ denotes the Riemannian connection of (M, g).

Note that since S has no engenvalue equal to 1 (s_{x} has only isolated fixed point), (I − S)^{−1} exists.

˜∇ is always complete.

Each tensor field P on M which is invariant by all s_{x} is parallel with respect to ˜∇, i.e., ˜∇P = 0. Therefore, the torsion tensor field T̃ and the curvature tensor field R̃ of ˜∇ are parallel:

˜∇T̃ = 0, ˜∇R̃ = 0.(∗)

Remark that if M is a symmetric space with standard {s_{x}}, ˜∇ = ∇.

Because of (∗), we have

˜∇R̃ = ˜∇T̃ = ˜∇S = ˜∇g = 0.

Therefore we can linearize Riemannian manifolds ((M, g), {s_{x} | x ∈ M}) with specific regular s-structure. We get algebraic objects (V, <, >, S, R̃, T̃), where S is a nonsingular linear isometry without fixed directions and R̃, T̃ are tensors of types (1, 3), (1, 2) respectively satisfying standard algebraic identities for the curvature and torsion. These objects are called infinitesimal models.

We have one-to-one correspondence between the set of simply connected Riemannian manifolds with regular s-structures and the set of infinitesimal models. Using the methods of linear algebra, we can classify regular s-structures and hence also generalized symmetric spaces (at least locally and for small dimensions).

If (M, g) is a generalized symmetric space and simply connected, then we have the de Rham decomposition

M = M_{1} × ⋯ × M_{r},

here if {s_{x}} is an s-tructure of order k of M, each M_{i} is a generalized symmetric space of order k_{i} and k_{i} divides k (i = 1, ⋯, r).

4 Theory of eigenvalues for generalized symmetric spaces

This theory is useful for the classification procedure of generalized symmetric spaces.

Let {s_{x} | x ∈ M} be a regular s-structure. Then S_{x} : T_{x}M ⟶ T_{x}M has the same eigenvalues for ∀x ∈ M. S_{x} is real, orthogonal, without fixed vectors. Hence its system (θ_{1}, ⋯, θ_{n}) of eigenvalues must satisfy

|θ_{i}| = 1, θ_{i} ≠ 1 for i = 1, ⋯, n

and

(θ̄_{1}, ⋯, θ̄_{n}) = (θ_{1}, ⋯, θ_{n}) up to numeration.

Every n-tuple with these properties is called an admissible n-tuple. Denote by P_{n} the set of all admissible n-tuples.

We now make P_{n} a partially ordered set. We introduce characteristic equations (of two classes):

a) X_{i}X_{j} = 1, i, j = 1, ⋯, n,

b) X_{i}X_{j} = X_{k}, i ≠ j ≠ k ≠ i, i, j, k = 1, ⋯, n.

Each element (θ_{1}, ⋯, θ_{n}) ∈ P_{n} satisfies at least one equation of type a) (n ≥ 2). (If the eigenvalues of S_{x} on (M, g) do not satisfy any equation of type b), then (M, g) is locally symmetric.)

We introduce the partial ordering ”⪯” on P_{n} as follows: (θ_{i}) ⪯ (θ′_{i}) if all characteristic equations satisfied by (θ_{i}) are also satisfied by (θ′_{i}) after possible re-numeration.

(Here (θ_{i}) ⪯ (θ′_{i}) and (θ′_{i}) ⪯ (θ_{i}) is possible, e.g., if (θ′_{i}) is a renumeration of (θ_{i}).)

Proposition 4.1. Let {s_{x} | x ∈ M} be a regular s-structure on (M, g) with the corresponding eigenvalues (θ_{i}). Let (θ′_{i}) ∈ P_{n} and (θ′_{i}) ⪰ (θ_{i}). Then there is a regular s-structure {s′_{x} | x ∈ M} on (M, g) with the eigenvalues (θ′_{i}).

Theorem 4.1. In the partially ordered set (P_{n}, ⪯), there is only a finite set of maximal elements, which are all of finite order.

[figure: ⟵ maximal elements]

This set will be denoted by D_{n}.

The same procedure works without regularity condition.

We only have to add new type of characteristic equations

c) X_{i}X_{j}X_{k} = X_{l}. The corresponding set D′_{n} of maximal elements contains D_{n}, but it is bigger.

Now we go back to the regular s-structures. We have the following theorem.

Theorem 4.2. For every dimension n ≥ 2, there is a finite set D_{n} = {(θ_{1}^{α}, ⋯, θ_{n}^{α}) | α = 1, ⋯, r(n)} with the following properties:

a) all elements of D_{n} are of finite order,

b) if (M, g) admits a regular s-structure, then it also admits a regular s-structure with the system of eigenvalues contained in D_{n}.

(Warning: the same (M, g) can admit regular s-structures corresponding to more elements of D_{n}).

This theorem is important for the classification of generalized symmetric spaces in low dimensions.

4.1 Basic systems of eigenvalues in small dimensions

D_{2} : (−1, −1)  (symmetric space)
D_{3} : (−1, −1, −1)  (symmetric space)
(i, −i, −1)  (4-symmetric space of dimension 3)
D_{4} : (−1, −1, −1, −1)  (symmetric space)
(θ, θ, θ^{2}, θ^{2}), θ = e^{\frac{2πi}{3}}  (∃ new example of order 3)
(i, −i, −1, −1)  (direct product or space of order 3)
(θ, θ^{2}, θ^{3}, θ^{4}), θ = e^{\frac{2πi}{5}}  (only space of order 3)
D_{5} : (−1, −1, −1, −1, −1)  (symmetric space)
(i, −i, −1, −1, −1)  (direct product)
(i, −i, i, −i, −1)  (∃ new examples of order 4)
(θ, θ^{2}, θ^{3}, θ^{4}, θ^{5}), θ = e^{\frac{2πi}{6}}  (∃ new examples of order 6)
(e^{\frac{πi}{4}} , e^{− \frac{πi}{4}} , i, −i, −1)  (no additional examples)

An estimate: Let k(n) denote the maximum of all orders of the elements in D_{n}. Then

k(n) ≤ 5^{\frac{n}{4}}  n : even,
k(n) ≤ 2 · 5^{\frac{n−1}{4}}  n : odd.

We see that k(4) ≤ 5 (exact estimate) and k(5) ≤ 10 (not exact, in fact k(5) ≤ 8). It is a natural question whether there is an estimate to improve the above inequality or not.

4.2 Classification in dimension n = 4

All generalized symmetric spaces of dimension 4 are symmetric spaces or spaces of order 3 isometric to one of the following forms:

M = ∥ \begin{matrix}a & b & u \\ c & d & v \\ 0 & 0 & 1 \end{matrix} ∥ / ∥ \begin{matrix}cos t & − sin t & 0 \\ sin t & cos t & 0 \\ 0 & 0 & 1 \end{matrix} ∥

where det ∥ \begin{matrix}a & b \\ c & d \end{matrix} ∥ = 1

M = (group of all equiaffine trans- formations of the Euclidean plane) / \binom{subgroup of all rotations ar-}{round the origin}

M ≅ ℝ^{4}[x, y, u, v] and then

g = (−x + \sqrt{x^{2} + y^{2} + 1})du^{2} + (x + \sqrt{x^{2} + y^{2} + 1})dv^{2}

−2ydudv + λ^{2} [\frac{(1 + y^{2})dx^{2} + (1 + x^{2})dy^{2} − 2xydxdy}{1 + x^{2} + y^{2}} ]

where λ > 0 is a parameter.

A typical symmetry of order 3 at the origin is given by

u′ = cos \frac{2π}{3} · u − sin \frac{2π}{3} · v,
v′ = sin \frac{4π}{3} · u + cos \frac{4π}{3} · v,
x′ = cos \frac{2π}{3} · x − sin \frac{2π}{3} · y,
y′ = sin \frac{4π}{3} · x + cos \frac{4π}{3} · y.

4.3 Classification in dimension n = 5

A generalized symmetric space of dimension 5 is locally isometric to one of the followings:

• a symmetric space,

• 11 families of order 4,

• 1 family of order 6.

4.3.1 Examples of order 4 – selection:

Example 1. Nilpotent matrix group

∥ \begin{matrix}1 & 0 & 0 & x \\ 0 & 1 & 0 & y \\ u & v & 1 & z \\ 0 & 0 & 0 & 1 \end{matrix} ∥ ≅ ℝ^{5}[x, y, z, u, v],

g = dx^{2} + dy^{2} + du^{2} + dv^{2} + λ^{2}(xdu − ydv + dz)^{2}, where λ > 0 arbitrary parameter.

Typical symmetry of order 4 at the origin is given by

x′ = −y, y′ = x, z′ = −z, u′ = −v, v′ = u.

Example 2. M = SO(3,)/SO(2) with the invariant Riemannian metrics depending on 3 real parameters. The symmetry at the origin of M is as follows

∥ \begin{matrix}a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{matrix} ∥ ↦ ∥ \begin{matrix}b̄_{2} & −b̄_{1} & b̄_{3} \\ −ā_{2} & ā_{1} & −ā_{3} \\ c̄_{2} & −c̄_{1} & c̄_{3} \end{matrix} ∥ .

Example 3. Solvable complex matrix group

M = ∥ \begin{matrix}e^{λt} & 0 & z \\ 0 & e^{−λt} & w \\ 0 & 0 & 1 \end{matrix} ∥ ≅ ℝ^{2}[z, w] × [t]

where z, w: complex variables, t: real variable, λ: complex parameter, λ ≠ 0.

A family of invariant metrics g such that (M, g) is irreducible (i.e., not a product of Riemannian manifolds).

Symmetry at the origin is given by

z′ = iw, w′ = iz, t′ = −t.

Example 4. M = SO(3) × SO(3)/SO(2)

SO(2) = \{(∥ \begin{matrix}cos t & − sin t & 0 \\ sin t & cos t & 0 \\ 0 & 0 & 1 \end{matrix} ∥ , ∥ \begin{matrix}cos t & sin t & 0 \\ − sin t & cos t & 0 \\ 0 & 0 & 1 \end{matrix} ∥) , t ∈ ℝ\}

Symmetry of order 4 at the origin is induced by the following automorphisms of the group GL(3,) × GL(3,):

(∥ \begin{matrix}a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{matrix} ∥ , ∥ \begin{matrix}ã_{1} & ã_{2} & ã_{3} \\ b̃_{1} & b̃_{2} & b̃_{3} \\ c̃_{1} & c̃_{2} & c̃_{3} \end{matrix} ∥) ↦ (∥ \begin{matrix}ã_{1} & −ã_{2} & −ã_{3} \\ −b̃_{1} & b̃_{2} & b̃_{3} \\ −c̃_{1} & c̃_{2} & c̃_{3} \end{matrix} ∥ , ∥ \begin{matrix}a_{1} & −a_{2} & a_{3} \\ −b_{1} & b_{2} & −b_{3} \\ c_{1} & −c_{2} & c_{3} \end{matrix} ∥)

Example 5. M = SU(3)/SU(2) ≅ S^{5} ((M, g) is not a standard sphere). For a special choice of the invariant metric, S^{5} has the metric induced from the Fubini–Study metric in P^{3} as a geodesic sphere in P^{3}.

4.3.2 A family of spaces of order 6

M = ∥ \begin{matrix}e^{−(u+v)} & 0 & 0 & x \\ 0 & e^{u} & 0 & y \\ 0 & 0 & e^{v} & z \\ 0 & 0 & 0 & 1 \end{matrix} ∥ ≅ ℝ^{5}[x, y, z, u, v] (A solvable Lie group)

We consider invarinat Riemannian metrics depending on two real parameters a > 0, b > 0:

g = a^{2}(du^{2} + dv^{2} + dudv) + (b^{2} + 1)(e^{2(u+v)}dx^{2} + e^{−2u}dy^{2} + e^{−2v}dz^{2})
+(b^{2} − 2)(e^{v}dxdy + e^{u}dxdz − e^{−(u+v)}dydz).

The typical symmetry at the origin is given by

x′ = y, y′ = −z, z′ = x, u′ = v, v′ = −(u + v).

This symmetry is of order 6!

The isotropy group of isometries at the origin consists of 8 elements.

Remark. Let (M, g) be not necessarily homogeneous in general. Let there exist a point p ∈ M such that p is an isolated fixed point for some isometry ∃s_{p} : M ⟶ M. Then there exists an isometry s′_{p} : M ⟶ M of finite order k:

k ≤ (4 + s(p))^{\frac{dimM+1}{2}}

where s(p) denotes the Singer number defined as follows: Put

G_{p}^{(l)} = {A ∈ Aut(T_{p}M) | A(g_{p}) = g_{p}, A(R_{p}) = R_{p}, ⋯, A((∇^{(l)}R)_{p}) = (∇^{(l)}R)_{p}},

l = 0, 1, 2, · · ·. Then there exists the minimal k ≥ 0 such that the sequence {G_{p}^{(l)}} stabilizes, i.e.,

G_{p}^{(0)} ⊃ G_{p}^{(1)} ⊃ ⋯ ⊃ G_{p}^{(k)} = G_{p}^{(k+1)} = ⋯

Such a number k = s(p) is called the Singer number.

[1] A. Gray, Riemannian manifolds with geodesic symmetries of order 3, J. Differential Geometry, 7(1972), 343–369.

[2] J.A. Jiménez, Riemannian 4-symmetric spaces, Trans. Amer. Math. Soc., 306(1988), 715–734.

[3] O. Kowalski, Generalized symmetric spaces, Lecture Notes in Mathematics, 805, Springer-Verlag, Berlin-New York, 1980.

[4] R. A. Marinosci, Classification of five-dimensional generalized pointwise symmetric Riemannian spaces, Geom. Dedicata, 57(1995), 11–53.

Oldřich Kowalski Faculty of Mathematics and Physics Charles University Sokolovská 83 186 00 Praha CZECH REPUBLIC e-mail: kowalski@karlin.mff.cuni.cz

URL: http://www.karlin.mff.cuni.cz/~kowalski/

COHOMOLOGY OF FLAT CONNECTIONS

JAN KUBARSKI

Institute of Mathematics, Technical University of Łódź aleja Politechniki 11, 90-924 Łódź, POLAND e-mail: kubarski@ck-sg.p.lodz.pl

April 26, 2001

Abstract

We study flat connections in spherical Lie algebroids (over oriented compact manifolds) defined everywhere but a finite number of points. Under some assumptions concerning dimensions with any such isolated singularity we join a real number called an index. For ℝ-spherical Lie algebroids, this index cannot be an integer. We prove the index theorem saying that the index sum is independent of the choice of a connection. Multiplying this index sum by the orientation class of M, we get the Euler class of this Lie algebroid. Some integral formulae for indexes are given.

1 Introduction

Lie algebroids arise in many subjects of differential geometry and play a role analogous to that of Lie algebras for Lie groups (i.e. compose an infinitesimal invariant). For example, they arise in the theory of differential groupoids, principal bundles and vector bundles [ P], [ L2], [ K-S], [ M1], [ K2], [ N1], [ N2], transversally complete and transversally parallelizable foliations [ MO1], [ MO2], nonclosed Lie subgroups [ MO2], [ K4], Poisson manifolds [ C-D-W], [ D-S], [ G1], [ V1], [ V2], [ Ko1], [ Ko2] and others (see the survey article by K.Mackenzie [M2], and also see J.Kubarski [ K6]). Connections – splittings of the Atiyah sequences of Lie algebroids (in regular case) – correspond in the majority of the above categories to well known geometric objects such as distributions or differential systems.

On the ground of Lie algebroids, we observe an interesting analogy of the theory of sphere bundles, namely, it turns out that, in some sense, the roles of flat connections for Lie algebroids and of cross-sections for sphere bundles correspond mutually. The common ideas are the index at a singularity and the theorem of Euler-Poincaré-Hopf type as well as some technique methods. This analogy was first noticed for Lie algebroids with 1-dimensional isotropy Lie algebras in the geometry of regular Poisson manifolds over ℝ-Lie foliations [ K8]. The main purpose of our work is to research this phenomenon in the domain of transitive Lie algebroids without a restriction of the dimensions of isotropy Lie algebras.

This paper is based on [ K7] and [ K10]. In [ K7] the idea of the fibre integral is adopted to regular Lie algebroids: the integration operator over the adjoint bundle of Lie algebras is defined, the class of Lie algebroids for which this operator commutes with differentials (giving then a homomorphism on cohomology) is characterized and many families of examples coming from principal bundles, TC-foliations and Poisson manifolds are given. Paper [ K10] deals with a subclass of the above class which contains so-called s-Lie algebroids, defined as the transitive ones with spherical isotropy Lie algebras i.e. cohomologically looking like a sphere (Lie algebras ℝ, sk (3, ℝ), sl (2, ℝ) are examples). For an s-Lie algebroid there is constructed a long exact sequence of cohomology (Gysin types) and the Euler class.

In this paper the cohomology theory of flat connections in s-Lie algebroids is developed. If the dimension of the base manifold is equal to n + 1, where n is the dimension of the isotropy Lie algebras, the index at a isolated singularity is defined. A version of the Euler-Poincaré-Hopf theorem joining the sum of indexes to the Euler class is given. In the context of Lie algebroids coming from S^{1} or Spin (3) principal bundles (over M^{2} or M^{4}, respectively) the above theorem generalizes the classical E-P-H theorem since cross-sections of these bundles determine flat connections (but not every connection determines a crosssection). In the end, some integral formulae for indexes are obtained. We add that in general the index can not be an integer and the sum of indexes has nothing in common with the Euler-Poincaré characteristic of the Lie algebroid understood as an alternative sum of the dimensions of the cohomology groups of this Lie algebroid.

2 Fibre integral and Gysin sequence

By a Lie algebroid on a manifold M [ P] we mean a system A = (A,

[[·, ·]], γ) consisting of a vector bundle A on M and mappings [[·, ·]] : SecA×SecA → SecA, γ : A → T M, such that (1) (SecA, [[·, ·]]) is an ℝ-Lie algebra, (2) γ, called the anchor, is a homomorphism of vector bundles, (3) Secγ : SecA → 𝔛(M), ξ ↦ γ ∘ ξ, is a homomorphism of Lie algebras, (4) [[ξ, f · η]] = f · [[ξ, η]] + (γ ∘ ξ)(f) · η, ξ, η ∈ SecA, f ∈ C^{∞}(M). The properties make the space of global cross-sections SecA a Lie-Rinehart algebra [called also a Lie module, a Lie pseudoalgebra, etc, according to particular authors] over the commutative algebra C^{∞} (M) [ H1]. A Lie algebroid A is said to be transitive if γ is an epimorphism of vector bundles, and regular if γ is of constant rank. When the isotropy Lie algebras 𝔤_{|x} of A are isomorphic to a given Lie algebra 𝔤 then A is shortly called a 𝔤-Lie algebroid. In the sequel, the notions and the notations from [ P], [ M1], [ K3], [ M-R], [ K1], [ M1] are adopted. Among them, the adjoint bundle of Lie algebras ggg := Kerγ, the Atiyah sequence 0 → ggg → A ⟶^{γ} F → 0, the notions of representations, connections, homomorphisms (strong or non-strong) of Lie algebroids as well as the exterior derivative d_{A} and the pullback of A-differential forms, are used.

In [ K7] we introduce the notion of a vertically oriented Lie algebroid as a pair (A, ε) consisting of a regular Lie agebroid A and a non-singular cross-section ε of ⋀^{n} ggg, n = rankggg. By a homomorphism of vertically oriented Lie algebroids (A, ε) → (A′, ε′) , assuming rankggg = rankggg′ = n, we mean a non-strong, in general, homomorphism T : A → A′, inducing t : M → M′, of Lie algebroids, such that (⋀^{n} T^{+}) (ε_{x}) = ε′_{tx}, x ∈ M (where T^{+} : ggg → ggg′ is the restriction of T to adjoint bundles). We write (T, t) : (A, ε) → (A′, ε′).

An operator of the fibre integral ∫̸_{A} in a vertically oriented Lie algebroid (A, ε) is introduced. For a transitive Lie algebroid (the case considered in this work), the fibre integral ∫̸_{A} : Ω_{A}^{⋆} (M) → Ω^{⋆−n} (M) is defined in the following way: ∫̸_{A}Φ = 0 if deg Φ < n and γ^{⋆} (∫̸_{A}Φ) = (−1)^{nk} ι_{ε}Φ if deg Φ = n + k, k ⩾ 0. We recall that Ω_{A} (M) denotes the space of real A-differential forms, i.e. the space Sec ⋀ A^{⋆} of cross-sections of the bundle ⋀ A^{⋆}. The following are basic properties (a) ∫̸_{A} ∘ γ^{⋆} = 0, (b) ∫̸_{A}γ^{⋆}ψ ∧ Φ = ψ ∧ ∫̸_{A}Φ for arbitrary forms ψ ∈ Ω (M) and Φ ∈ Ω_{A} (M) , (c) ∫̸_{A} is an epimorphism [ K7].

The operator ∫̸_{A} commutes with the exterior derivatives d_{A} and d_{M} if and only if (a1) the isotropy Lie algebras ggg_{∣x} are unimodular, and (a2) the cross-section ε is invariant with respect to the adjoint representation of A on ⋀^{n} ggg. The transitive Lie algebroid A fulfilling properties (a1) and (a2) above is shortly called a TUIO-Lie algebroid. In [ K7] and [ K9] many examples can be found. The paper [ K10] deal with the subcategory of TUIO-Lie algebroids for which isotropy Lie algebras 𝔤 are spherical, i.e. satisfy conditions H^{k} (𝔤) = ℝ for k = 0, dim 𝔤, and H^{k} (𝔤) = 0 for 1 ≤ k ≤ dim 𝔤 − 1. Such Lie algebroids are called briefly s-Lie algebroids. Many examples for principal bundles and TC-foliations are given. For an s-Lie algebroid (A, ε) , there is constructed a long exact sequence of cohomology (Gysin sequence)

⋯ ⟶ H^{p} (M) ⟶^{D^{p}} H^{p+n+1} (M) ⟶^{γ^{#}} H_{A}^{p+n+1} (M) \underrightarrow{∫̸_{A}^{#}} H^{p+1} (M) ⟶^{D^{p+1}} ⋯  (2.2.1)

The class χ_{(A,ε)} := D^{0} (1) ∈ H^{n+1} (M) is called the Euler class of (A, ε) and χ_{(A,ε)} = [Ψ] where γ^{∗}Ψ = d_{A}Φ for ∫̸_{A}Φ = −1. The equality D (α) = α ∧ χ_{(A,ε)} holds. The Euler class χ_{(A,ε)} can be computed via the Chern-Weil homomorphism of A (introduced in [ K3]).

3 Difference class

By a connection in a transitive Lie algebroid A we mean a splitting λ : T M → A of the Atiyah sequence 0 → ggg ↪ A →^{γ} T M → 0, i.e. a homomorphism of vector bundles such that γ ∘ λ = id. If λ is a homomorphism of Lie algebroids λ ∘ [X, Y] = [[λ ∘ X, λ ∘ Y]], X, Y ∈ 𝔛 (M) , then λ is called flat. In this situation, the pullback of differential forms λ^{⋆} : Ω_{A} (M) → Ω (M) commutes with differentials λ^{⋆} ∘ d_{A} = d_{M} ∘ λ^{⋆}, giving — on cohomology — a homomorphism of algebras λ^{#} : H_{A} (M) → H (M) . Let λ be a flat connection in an s-Lie algebroid (A, ε). According to the exactness of sequence (2.2.1), [ K7, Prop. 4.2.1 (b)] and [ K10, Cor. 3.4] we can easily show that

H_{A} (M) = ker∫̸_{A}^{#} ⨁ ker λ^{#},  (3.3.1)

∫̸_{A}^{#}| ker λ^{#} : ker λ^{#} ⟶^{≅} H (M) .  (3.3.2)

By the above, there exists a uniquely determined cohomology class

ω_{λ} ∈ ker λ^{#n} ⊂ H_{A}^{n} (M)

such that ∫̸_{A}^{#}ω_{λ} = 1 ∈ H^{0} (M). ω_{λ} depends on the mapping λ^{#} : H_{A} (M) → H (M) only. The class ω_{λ} is called the cohomology class of a flat connection λ.

For two flat connections λ, σ : T M → A, their cohomology classes ω_{λ}, ω_{σ} ∈ H_{A}^{n} (M) satisfy the equality ∫̸_{A}^{#} (ω_{λ} − ω_{σ}) = 0. Thanks to the exactness of sequence (2.2.1) there exists a cohomology class [λ, σ] ∈ H^{n} (M) such that

ω_{λ} − ω_{σ} = γ^{#} [λ, σ] .

Definition 3.1. The class [λ, σ] is called the difference class of flat connections λ and σ in the s-Lie algebroid (A, ε) .

Proposition 3.2. For flat connections λ and σ in an s-Lie algebroid (A, ε) , we have λ^{#}α− σ^{#}α = − (∫̸_{A}^{#}α) ∧ [λ, σ] , α ∈ H_{A} (M) .

Since γ^{#} is a monomorphism for a flat Lie algebroid A, we obtain

Corollary 3.3. Let λ and σ be two flat connections in an s-Lie algebroid A. Then the following conditions are equivalent: (a) λ^{#} = σ^{#}, (b) ω_{λ} = ω_{σ}, (c) [λ, σ] = 0. If H^{n} (M) = 0, then, clearly, H^{n} (M) ∋ [λ, σ] = 0, therefore λ^{#} = σ^{#}.

Lemma 3.4. Let (A, ε) be an arbitrary s-Lie algebroid with rankggg = n. For a representative Ψ of the Euler class χ_{(A,ε)} and an n-form Φ ∈ Ω_{A}^{n} (M) such that ∫̸_{A}Φ = −1 and d_{A}Φ = γ^{⋆}Ψ, for any open subset U ⊂ M and two flat connections λ, σ : T U → A_{|U}, the following equalities hold:

(1) ω_{σ} = [γ_{|U}^{⋆}σ^{⋆} (Φ_{|U}) − Φ_{|U}],

(2) [λ, σ] = [λ^{⋆}Φ_{|U} − σ^{⋆}Φ_{|U}].

Theorem 3.5 (The naturality of the difference class). Let (T, t) : (A, ε) → (A′, ε′) be a homomorphism of s-Lie algebroids such that T_{x} : A_{∣x} → A′_{∣tx}, x ∈ M, are isomorphisms.

(a) Assume that σ, σ′ are flat connections in A and A′, respectively, such that T ∘σ = σ′∘t_{∗}. Then T^{#}ω_{σ′} = ω_{σ}.

(b) For any two pairs of such flat connections (λ, λ′) , (σ, σ′) , we obtain t^{#}([λ′, σ′]) = [λ, σ] .

Proof. (a): To check (a), it is sufficient to notice that σ^{#} (T^{#}ω_{σ′}) = (T ∘ σ)^{#} ω_{σ′} = (σ′ ∘ t_{⋆})^{#} ω_{σ′} = t^{#}σ^{′#}ω_{σ′} = 0 and (see [ K7]) ∫̸_{A}^{#}T^{#}ω_{σ′} = t^{#}∫̸_{A′}^{#}ω_{σ′} = 1.

(b): According to the fact that γ^{#} is a monomorphism, we only need to notice

γ^{#}t^{#} [λ′, σ′] = (t_{⋆} ∘ γ)^{#} [λ′, σ′] = (γ′ ∘ T )^{#} ([λ′, σ′]) = T^{#} (ω_{λ′} − ω_{σ′})
= ω_{λ} − ω_{σ} = γ^{#} [λ, σ] .

The folowing theorem gives a relationship between the Euler class and the difference class (compare with the classical theorem for sphere bundles, for example [1]).

Theorem 3.6. Let {U, V } be an open covering of M and let λ_{U} : T U → A_{|U} and σ_{V} : T V → A_{|V} be flat connections in (A, ε) over U and V, respectively (U, V need not be connected). Consider the Mayer-Vietoris sequence of the triad {M, U, V } for the usual real de Rham cohomology and let ∂ : H (U ∩ V) → H (M) be the connecting homomorphism. Then

χ_{(A,ε)} = ∂ [λ, σ]

where λ = λ_{U}|_{U∩V} and σ = σ_{V} |_{U∩V} .

Proof. For the inclusions j_{1} : U ∩ V ↪ U and j_{2} : U ∩ V ↪ V according to Lemma 3.4, [λ, σ] = [λ^{⋆}Φ_{|U∩V} − σ^{⋆}Φ_{|U∩V} ] = [j_{1}^{⋆}(λ_{U}^{⋆}Φ_{|U})−j_{2}^{⋆}(σ_{V}^{⋆} Φ_{|V})]. Since d(λ_{U}^{⋆}Φ_{|U}) = λ_{U}^{⋆}d_{A_{|U}} Φ_{|U} = λ_{U}^{⋆}γ_{|U}^{⋆}Ψ_{|U} = Ψ_{|U}, analogously d(σ_{V}^{⋆} Φ_{|V}) = Ψ_{|V} , we get — via the construction of ∂ — ∂ [λ, σ] = [Ψ] = χ_{(A,ε)}.

4 The index of a flat connection at an isolated singular point and the Euler number

By a local connection with singularity at a point a ∈ M in a Lie algebroid A we mean the connection

σ : T U̇ → A_{|U̇}, a ∈ U ⊂ M (U is open), U̇ = U \ {a} .

Let (A, ε) be an arbitrary s-Lie algebroid over an n + 1-dimensional oriented manifold M (n ⩾ 1) with n = rankggg and let σ : T U̇ → A_{|U̇} be a local connection with singularity at a ∈ U ⊂ M. Take additionally a neighbourhood V ∋ a such that V ⊂ U and V ≅ ℝ^{n+1}. A_{|V} possesses, [ M1], a global flat connection λ : T V → A_{|V} . Denote λ|V̇ (V̇ = V \{a}) by λ̇

and consider the difference class [λ̇, σ|V̇] ∈ H^{n}(V̇). Let α_{V} : H^{n}(V̇) ⟶^{≅} ℝ be the canonical mapping [1, Vol.I] (V̇ has the orientation induced from M). By analogous reasoning as in the theory of sphere bundles [1, Vol.I] and due to Corollary 3.3 we check that the number α_{V} ([λ̇, σ|V̇]) is independent of the auxiliary flat connection λ and of the neighbourhood V. This means α_{V} ([λ̇, σ|V̇]) depends only on the choice of σ.

Definition 4.1. The number α_{V} ([λ̇, σ|V̇]) is called the index of σ at a and denoted by

j_{a} (σ) .

Proposition 4.2 (Naturality of the index). Let (Â, ε̂) be another s-Lie algebroid over an oriented n + 1-dimensional manifold M̂ and (T, t) : (Â, ε̂) → (A, ε) be a homomorphism of s-Lie algebroids fulfilling conditions

T_{x} : Â_{∣x} → A_{∣tx}, x ∈ M, is an isomorphism,

t : M̂ → M is a diffeomorphism onto an open subset.

Let a ∈ M, â ∈ M̂, t \binom{}{â} = a. Take a local flat connection σ : T U̇ → A_{|U̇} with singularity at a. Then the mapping T^{#}σ : T Ẇ → Â_{|Ẇ} , W = t^{−1} [U] , Ẇ = W \ \left\{{ \atop â}\right\} , defined by (T^{#}σ) (v) = T_{|pv}^{−1} (σ (t_{⋆}v)) is a flat connection in Â with singularity at â, and jâ (T^{#}σ) = j_{a} (σ) .

The main goal of this article is a theorem joining the index sum ∑ j_{a_{v}} (σ) of any flat connection with a finite number of singularities {a_{1}, …, a_{k}} to the Euler class of Lie algebroid.

Theorem 4.3 (The Euler-Poincaré-Hopf theorem for flat connections). Let (A, ε) be an s-Lie algebroid of rank n over an oriented compact manifold M of dimension n+1 and let σ : T (M \ {a_{1}, …, a_{k}}) → A be a flat connection with singularities at points a_{1}, …, a_{k}. Then the Euler class χ_{(A,ε)} ∈ H^{n+1} (M) is given by the formula

χ_{(A,ε)} = (∑_{v=1}^{k} j_{a_{v}} (σ)) · ω_{M}

where ω_{M} is the orientation class of M; equivalently, ∫_{M}^{#} χ_{(A,ε)} = ∑_{v=1}^{k} j_{a_{v}} (σ). In particular, the index sum ∑_{v=1}^{k} j_{a_{v}} (σ) is independent of the choice of the connection.

Proof. For each v = 1, …, k, choose a neighbourhood U_{v} ∋ a_{v} diffeomorphic to ℝ^{n+1} and such that the sets U_{v} are pairwise disjoint. Put U = ⋃ U_{v}, V = M \ {a, …, a_{k}} . Then M = U ∪ V and U ∩ V = ⋃ U̇_{v} where U̇_{v} = U_{v} \ {a_{v}} . Take arbitrary flat connections λ̃_{v} : T U_{v} → A_{|U_{v}}, v = 1, …, k. The family {λ̃_{v}} determines one flat connection λ̃ : T U → A_{|U} such that λ̃|_{U_{v}} = λ̃_{v}. Define λ̌ = λ̃|_{U∩V} and σ̌ = σ|_{U∩V} . According to Theorem 3.6, χ_{(A,ε)} = ∂[λ̌, σ̌]. In the sequel, put λ_{v} = λ̃_{v}|U̇_{v} and σ_{v} = σ|U̇_{v}. Then [λ̌, σ̌] = ⊕_{v} [λ_{v}, σ_{v}] . By [1, Prop.VII Chap.VI, Vol.I] ∫_{M}^{#} ∘∂ = α, where α : ⨁_{v} H^{n}(U̇_{v}) → ℝ is equal to ⊕β_{v} ↦ ∑ α_{U_{v}} (β_{v}) . Therefore we get

∫_{M}^{#} χ_{(A,ε)}_{A} = ∫_{M}^{#} ∂[λ̌, σ̌] = ∫_{M}^{#} ∂ (⊕_{ν} [λ_{v}, σ_{v}]) = α (⊕_{ν} [λ_{v}, σ_{v}])

= ∑_{v=1}^{k} α_{U_{v}} [λ_{v}, σ_{v}] = ∑_{v=1}^{k} j_{a_{v}} (σ) .

The sum

j_{(A,ε)} = ∑_{v=1}^{k} j_{a_{v}} (σ)

is called the Euler number of the s-Lie algebroid (A, ε) . According to Theorem 5.4 from [ K10], the Euler number j_{(A,ε)} is not – in general – an invariant of the cohomology algebra of A and has nothing in common with the Euler-Poincaré characteristic of A. The last, when considered for TUIO-Lie algebroids (dim M + rankggg is odd), always is 0 [ K9].

5 Integral formulae

Proposition 5.1. For a trivial s-Lie algebroid A = T M × 𝔤 vertically oriented by a tensor 0 ≠ ε_{o} ∈ ⋀^{n} 𝔤 (n = dim 𝔤) and equipped with a ”standard” flat connection τ_{0} : T M → T M × 𝔤, v ↦ (v, 0) , we have

(a) if σ : T M → A is a flat connection, then its cohomology class ω_{σ} is given by ω_{σ} = −σ̂^{#} [φ_{o}] × 1 + 1 × [φ_{o}] where σ̂ = pr_{2} ∘ σ : T M → 𝔤 and φ_{o} ∈ ⋀^{n} 𝔤^{⋆} is a tensor such that ι_{ε_{o}}φ_{o} = 1. In particular ω_{τ_{0}} = 1 × [φ_{o}] .

(b) The difference class [τ_{0}, σ] is equal to [τ_{0}, σ] = σ̂^{#} [φ_{o}] .

Proof. Using the fact that the projection pr_{2} : T M → 𝔤 is a nonstrong homomorphism of Lie algebroids [ K5], we can easily see that

σ^{#} (−σ̂^{#} [φ_{o}] × 1 + 1 × [φ_{o}]) = 0,
∫̸_{A}^{#} (−σ̂^{#} [φ_{o}] × 1 + 1 × [φ_{o}]) = 1.

Now (a) follows from the definition of the cohomology class ω_{σ} whereas (b) may be obtained from the definition of the difference class and the equality

ω_{τ_{0}} − ω_{σ} = 1 × [φ_{o}] − (−σ̂^{#} [φ_{o}] × 1 + 1 × [φ_{o}]) = γ^{#}σ̂^{#} [φ_{o}] .

Corollary 5.2. For arbitrary flat connections λ, σ : T M → T M × 𝔤,

[λ, σ] = (σ̂^{#} − λ̂^{#}) [φ_{o}] .

Put ℝ^{.}^{n+1} = ℝ^{n+1} \ {0} and let 𝔤 be any n-dimensional unimodular Lie algebra 𝔤. Take tensors 0 ≠ ε_{o} ∈ ⋀^{n} 𝔤, φ_{o} ∈ ⋀^{n} 𝔤^{⋆} joined by the relation ι_{ε_{o}} (φ_{o}) = 1. Fix a flat connection σ : T ℝ^{.}^{n+1} → T ℝ^{.}^{n+1} × 𝔤 in the trivial Lie algebroid A = T ℝ^{n+1} × 𝔤 (oriented by the tensor ε_{o}). Let i : S^{n} ↪ ℝ^{.}^{n+1} be the inclusion.

Proposition 5.3. The index j_{0} (σ) of σ is given by the formula

j_{0} (σ) = ∫_{S^{n}} σ_{S}^{⋆} (φ_{o})  (5.5.1)

where σ_{S} is a nonstrong homomorphism of Lie algebroids defined as the composition σ_{S} :

T S^{n} ↪^{i_{⋆}} T ℝ^{.}^{n+1} ⟶^{σ̂} 𝔤.

Treat now σ̂ : T ℝ^{.}^{n+1} → 𝔤 as a 1-form on ℝ^{.}^{n+1} with values in 𝔤 and take the exterior

n-product σ̂ ∧…∧σ̂ ∈ Ω^{n}(ℝ^{.}^{n+1}; ⋀^{n} 𝔤). We have σ̂^{⋆} (φ_{o}) = \frac{1}{n!} (φ_{o})_{⋆} (σ̂ ∧ … ∧ σ̂) . Therefore

j_{0} (σ) = \frac{1}{n!}φ_{o} (∫_{S^{n}} i^{⋆} (σ̂ ∧ … ∧ σ̂)) .  (5.5.2)

Example 5.4. Each trivial Lie algebroid A = T ℝ^{n+1} × 𝔤 is integrable: A = A (P) for P = ℝ^{n+1} × G where G is an arbitrary Lie group with the Lie algebra 𝔤. A connection σ : T ℝ^{.}^{n+1} → A induces a connection H ⊂ T (ℝ^{.}^{n+1} × G) in the principal bundle ℝ^{.}^{n+1} × G, and the flatness of σ means the integrability of H. Assume a leaf L of the foliation H is the graph of some function f : ℝ^{.}^{n+1} → G. (If n ≥ 2, then such a function always exists which follows from the simple connectedness of ℝ^{.}^{n+1} and the reduction theorem [ K-N]). Therefore σ̂ (v) = R_{f(x)⋆}^{−1} (f_{⋆}v) , R_{f(x)} which is the right translation on G by f(x), and f^{⋆} (Δ_{R}) = ⟨φ_{o}, \frac{1}{n!} (σ̂ ∧ … ∧ σ̂)⟩ for the right-invariant n-form Δ_{R} ∈ Ω_{R}^{n} (G) equalling φ_{o} at the unity e of G.

(A) If G is compact, n-dimensional, oriented by Δ_{R} and the Lie algebra of G is spherical, then as a consequence of (5.5.1) and (5.5.2) we have

j_{0} (σ) = ∫_{S^{n}} (f_{|S^{n}})^{⋆} Δ_{R} = deg (f_{|S^{n}}) · ∫_{G} Δ_{R} .  (5.5.3)

As a corollary (taking any mapping f : ℝ^{.}^{n+1} → S^{n} such that f_{|S^{n}} = id_{S^{n}}), we obtain the existence of a local, flat singular connection having a nonzero index at the singularity.

Formula (5.5.3) yields that the set of real numbers being the indexes at a given point of singular local, flat connections coming from functions is discrete (more exactly, is equal to the set of multiples of ∫_{G} Δ_{R}). Such a situation takes place, for example, for all flat connections in any sk (3, ℝ)-Lie algebroid over M^{4} (since we can take G = SO(3)).

(B) If G is not compact, then Δ_{R} = d (Θ) for some Θ and

j_{0} (σ) = ∫_{S^{n}} (f_{|S^{n}})^{⋆} Δ_{R} = ∫_{S^{n}} d (f_{|S^{n}}^{∗}Θ) = 0.

Such a situation takes place, for example, in any sl (2, ℝ)-Lie algebroid over M^{4} (since we can take G = SL (2, ℝ)). Clearly, this fact can be noticed immediately by using a base e, f, g of sl (2, ℝ) such that [e, f] = g, [f, g] = 2f, [g, e] = 2e.

For an ℝ-Lie algebroid, not every local, singular and flat connection comes from a function, see the below example.

Example 5.5. In any ℝ-Lie algebroid over M^{2} we can construct a local, flat and singular connection whose index is a preassigned real number. Indeed, since ℝ is abelian, therefore the flatness of σ is equivalent to the closedness of the 1-form σ̂ on M^{2}. In this case, the product k · σ̂, k ∈ ℝ, also gives a flat connection. Therefore, if σ : T ℝ^{.}^{2} → T ℝ^{.}^{2} × ℝ, v ↦ (v, σ̂ (v)) , is a flat connection with a nonzero index at 0, j_{0} (σ) ≠ 0, then, for an arbitrary real number k ∈ ℝ, the mapping

τ : T ℝ^{.}^{2} ⟶ T ℝ^{.}^{2} × ℝ, v ⟼ (v, \frac{k}{j_{0} (σ)} · σ̂ (v)) ,

is a flat connection with j_{0} (τ) = k. Except for a discrete set of real numbers, this connection does not come from a function. More implicitly, considering ε_{0} = 1 ∈ ℝ and taking τ̂ = \frac{k}{2π} (\frac{x}{x^{2}+y^{2}} dy − \frac{y}{x^{2}+y^{2}} dx) , k ∈ ℝ, we have j_{0} (τ) = ∫_{S^{1}} τ̂ = k.

Example 5.6. In the Hopf S^{1}-bundle P = (S^{3} → S^{2}), for two different points p_{1}, p_{2} ∈ S^{2} and for any real number k ∈ ℝ, there exists a global flat connection σ_{k} with two singularities at {p_{1}, p_{2}} , such that the index j_{p_{1}} (σ_{k}) is equal to k. Indeed, since the Euler class of P is equal to the orientation class of S^{2}, any flat connection λ with a singularity at {p_{1}} has the index at p_{1} equal to 1 (assuming ∫_{G} Δ_{R} = 1). Take p_{2} ≠ p_{1} and M = S^{2} \ {p_{2}}. Since M is contractible, P_{|M} is trivial P_{|M} ≅ M × S^{1}. The connection λ determines a connection λ̄ : T (M\ {p_{1}}) → A ((M\ {p_{1}}) × S^{1}) = T (M\ {p_{1}}) × ℝ. Take σ̂ = pr_{2} ∘ λ̄ : T (M\ {p_{1}}) → ℝ. For an arbitrary real number k ∈ ℝ,

σ̄_{k} : T (M\ {p_{1}}) ⟶ T (M\ {p_{1}}) × ℝ, v ⟼ (v, k · σ̂ (v)),

is a flat connection. σ̄_{k} determines a flat connection σ_{k} in P with a singularity at {p_{1}, p_{2}} , such that j_{p_{1}} (σ_{k}) = k.

In the end we give some remarks concerning the existence of a connection with a finite number of singularities. We start with the case 𝔤 = ℝ.

Proposition 5.7. In each invariantly oriented ℝ-Lie algebroid over an arbitrary manifold M for which H^{2} (M) = 0 there exists a flat connection, in particular, when M is 2-dimensional non-compact.

Proof. According to [ K7], each invariantly oriented ℝ-Lie algebroid A over M is isomorphic to (M × ℝ) ⨁ T M with pr_{2} : (M × ℝ) ⨁ T M → T M as the anchor and the bracket [[·, ·]] defined via some closed real 2-form Ω in the following way: [[(f, X) , (g, Y)]] = (−Ω (X, Y) + ∂_{X}g − ∂_{Y} f, [X, Y]) , X, Y ∈ 𝔛 (M) , f, g ∈ C^{∞} (M). Each connection λ : T M → (M × ℝ) ⨁ T M has the form λ (v) = (λ̄ (v) , v) for a 1-differential form λ̄ ∈ Ω^{1} (M). A simple calculation shows that λ is flat if and only if d \binom{}{λ̄} = Ω. If H^{2} (M) = 0, such a 1-form exists.

As a corollary we get

Corollary 5.8. In each s-Lie algebroid of rank 1 over a compact 2-manifold M there exists a flat connection with a beforehand finite non-empty set of isolated singularities.

If a sk (3, R)-Lie algebroid over a compact 4-manifold comes from a Spin (3)-principal bundle, then - of course - it possesses a flat connection with one singularity (since such a cross-section of the sphere bundle exists [1, Vol. I]). In the general case, the problem is open.

The problem for sl (2, ℝ)-Lie algebroids looks differently. Namely, by the main theorem (4.3) and Example 5.4 (B) we have that χ_{(A,ε)} = 0 for any invariantly oriented sl (2, ℝ)-Lie algebroid (A, ε) over a compact connected oriented manifold M^{4}, admitting a flat connection with a finite number of isolated singularities. Really, such a Lie algebroid is flat since locally we can remove a singularity: if σ is a flat connection in T ℝ^{.}^{4} × sl (2, ℝ) then σ is given by a function f : ℝ^{.}^{4} → SL (2, ℝ). Using the fact that the third group of homotopy of SL (2, ℝ) is zero, π_{3} (SL (2, ℝ)) = 0, we can find f̄ : ℝ^{4} → SL (2, ℝ) such that f (x) = f̄(x) for ∥x∥ ≥ ε for a given small ε. This implies that we may remove the singularity at 0.

[ C-D-W] A.Coste, P.Dazord, A.Weinstein, Groupoides symplectiques, Publ. Dep. Math. Universite de Lyon 1, 2/A (1987).

[ D-S] P.Dazord, D.Sondaz, Varietes de Poisson — Algebroides de Lie, Publ. Dep. Math. Universite de Lyon 1, 1/B (1988).

[ G1] J. Grabowski, Lie algebroids and Poisson-Nijenhuis structures, Reports on Mathematics Physics, Vol. 40 (1997).

[ G-H-V] W. Greub, S. Halperin, R. Vanstone, Connections, Curvature, and Cohomology, Vol.I, 1971, Vol.II 1973, Vol.III 1976, New York and London.

[ H1] Huebschmann J, Poisson cohomology and quantization, J. reine angew. Math. 408, 1990, 57-113.

[ Ko1] Y. Kosmann-Schwarzbach, Exact Gerstenhaber Algebras and Lie Bialgebroids, Acta Applicandae Mathematicae 41: 153-165, 1995.

[ Ko2] —, The Lie Bialgebroid of a Poisson-Nijenhuis Manifold, Letters Mathematical Physics 38: 421-428, 1996.

[ K-N] S. Kobayashi, K. Nomizu,Foundations of differential geometry, Vol.I, Interscience Publishers, New York, London, 1963.

[ K1] J. Kubarski, Pradines-type groupoids over foliations; cohomology, connections and the Chern-Weil homomorphism, Preprint Nr 2, Institute of Mathematics, Technical University of Łódź, August 1986.

[ K2] —, Lie algebroid of a principal fibre bundle, Publ. Dep. Math. University de Lyon 1, 1/A, 1989.

[ K3] —, The Chern-Weil homomorphism of regular Lie algebroids, Publ. Dep. Math. University de Lyon 1, 1991.

[ K4] — , A criterion for the Minimal Closedness of the Lie Subalgebra Corresponding to a Connected Nonclosed Lie Subgroup, REVISTA MATEMATICA de la Universidad Complutense de Madrid, Vol.4, nu 2 y 3; 1991.

[ K5] —, Invariant cohomology of regular Lie algebroids, Proceedings of the VII International Colloquium on Differential Geometry, Spain 26-30 July, 1994, World Scientific, Singapure 1995.

[ K6] —, Bott’s Vanishing Theorem for Regular Lie Algebroids, Transaction of the A.M.S. Vol. 348, N 6, June 1996.

[ K7] —, Fibre integral in regular Lie algebroids, New Developments in Differential Geometry, Budapest 1996, Proceedings of the Conference on Differential Geometry, Budapest, Hungary, July 27-30, 1996; Kluwer Academic Publishers 1999.

[ K8] —, Connections in regular Poisson manifolds over ℝ-Lie foliations, Banach Center Publications, Vol 51, ”Poisson geometry”, Warszawa 2000, 141-149.

[ K9] —, Poincaré duality for transitive unimodular invariantly oriented Lie algebroids, Topology and Its Applications, in print.

[ K10] —, Gysin sequence and Euler class of spherical Lie algebroids, Publicationes Mathematicae Debrecen, in print.

[ K-S] A. Kumpera and D. C. Spencer, Lie equations, Vol. I: general theory, Annals of Math. Studies, No.73, Princeton University Press, Princeton, 1972.

[ L2] —, Sur les prolongements des fibrés principaux et groupoides différentiables banachiques, Seminaire de mathématiques superieures - élé 1969, Analyse Globale, Les presses de l’Université de Montréal, 1971, 7-108.

[ M1] K. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, Cambridge University Press, 124, 1987.

[ M2] —, Lie algebroids and Lie pseudoalgebras, Bull. London Math. Soc. 27 (1995), 97-147.

[ MO1] P.Molino, Etude des feuilletages transversalement complets et applications, Ann.Sci. Ecole Norm. Sup., 10(3) (1977), 289-307.

[ MO2] P.Molino, Riemannian Foliations, Progress in Mathematics Vol.73, Birkhäuser Boston Basel, 1988.

[ M-R] L.Maxim-Raileanu, Cohomology of Lie algebroids, An. Sti. Univ. ”Al. I. Cuza” Iasi. Sect. I a Mat. XXII f 2 (1976), 197-199.

[ N1] Ngo Van Que, Du prolongement des espaces fibrés et des structures infinitésimales, Ann. Inst. Fourier, Grenoble, 17.1, 1967, 157-223.

[ N2] —, Sur l’espace de prolongement différentiable, J. of. Diff. Geom., Vol.2, No.1, 1968, 33-40.

[ P] J. Pradines, Théorie de Lie pour les groupoides differentiables, Atti del Convegno Internazionale di Geometria Differenziale, Bologna, 28-30, IX, 1967.

[ V1] I. Vaisman, Complementary 2-forms of Poisson Strucures, preprint.

[ V2] —, The BV-algebra of a Jacobi manifolds, preprint.

CONFIGURATION SPACES AND ALGEBROIDS

VITALY KUSHNIREVITCH and ROMAN KADOBIANSKI

Abstract

Let M be complete, connected, oriented, C^{∞} (noncompact) Riemannian manifold of dimension d. The configuration space Γ_{M} over M is the set of locally finite subsets in M:

Γ_{M} := {γ ⊂ M : card (γ ∩ K) < ∞ for each compact K ⊂ M}.

Any γ ∈ Γ_{M} is identified with positive integer-valued Radon measure. The tangent space to Γ_{M} at a point γ is defined as Hilbert space T_{γ}Γ_{M} := L^{2}(M → T M; γ) (or, equivalently, T_{γ}Γ_{M} := ⨁_{x∈γ} T_{x}M). (See S.Albeverio, Yu.G.Kondratiev, M.Röckner, JFA 157 (1998).)

High order differential forms and de Rham cohomology on configuration spaces can also be considered (see S.Albeverio, A.Daletskii, E.Lytvynov, JGeomPhys, to appear). The main topic of discussion is to consider these objects from the Lie algebriods theory point of view.

Roman KADOBIANSKI National Technical University of Ukraine ”Kiev Polytechnic Institute” UKRAINE e-mail: tskafits@adam.kiev.ua

Vitaly KUSHNIREVITCH National Technical University of Ukraine ”Kiev Polytechnic Institute” UKRAINE e-mail: tskafits@adam.kiev.ua

EXTENDED FINITE CALCULUS – AN EXAMPLE OF ALGEGRAIZATION OF ANALYSIS

ANDRZEJ KRZYSZTOF KWAŚNIEWSKI

Institute of Computer Science, Białystok University PL – 15-887 Białystok, ul. Sosnowa 64, Poland e-mail: kwandr@noc.uw.edu.pl

Abstract “A Calculus of Sequences” started in 1936 by Ward constitutes the general scheme for extensions of classical operator calculus of Rota - Mullin considered by many afterwards and after Ward. Because of the notation we shall call the Wards calculus of sequences in its afterwards elaborated form - a ψ-calculus.

The ψ-calculus in parts appears to be almost automatic, natural extension of classical operator calculus of Rota - Mullin or equivalently - of umbral calculus of Roman and Rota.

At the same time this calculus is an example of the algebraization of the analysis - here restricted to the algebra of polynomials. Many of the results of ψ-calculus may be extended to Markowsky Q-umbral calculus where Q stands for a generalized difference operator i.e. the one lowering the degree of any polynomial by one.

The article is supplemented by the short indicatory glossaries of terms and notation used by Ward, Viskov, Markowsky, Roman on one side and the Rota-oriented notation on the other side.

KEY WORDS: extended umbral calculus , Graves-Heisenberg-Weyl algebra MCS (2000) : 05A40 , 81S99

1 Introduction

We shall call the Wards calculus of sequences [1] in its afterwards last century elaborated form - a ψ-calculus because of the notation [2]-[7]. The efficiency of the Rota oriented language and our notation used has been already exemplified by easy proving of ψ-extended counterparts of all representation independent statements of ψ-calculus [2]. Here these are ψ-labelled representations of Graves-Heisenberg-Weyl (GHW) algebra of linear operators acting on the algebra P of polynomials.

As a matter of fact ψ-calculus becomes in parts almost automatic extension of Rota - Mullin calculus or equivalently - of umbral calculus of Roman and Rota [8, 9, 10]. The ψ-extension relies on the notion of ∂_{ψ}-shift invariance of operators with ψ-derivatives ∂_{ψ} staying for equivalence classes representatives of special differential operators lowering degree of polynomials by one [6, 7, 11]. Many of the results of ψ-calculus may be extended to Markowsky Q-umbral calculus [11] where Q stands for arbitrary generalized difference operator i.e. the one lowering the degree of any polynomial by one. Q-umbral calculus [11] - as we call it - includes also those generalized difference operators, which are not series in ψ-derivative ∂_{ψ} whatever an admissible ψ sequence would be.

The note is at the same time the operator formulation of “A Calculus of Sequences” started in 1936 by Ward [1] with the indication of the role the ψ-representations of Graves-Heisenberg-Weyl (GHW) algebra in formulation and derivation of principal statements of the ψ-extension of finite operator calculus of Rota.

Restating what was said above we observe that all statements of standard finite operator calculus of Rota are valid also in the case of ψ-extension under the almost automatic replacement of {D, x̂, id} generators of GHW by their ψ-representation correspondents {∂_{ψ}, x̂_{ψ}, id} - see definitions 2.1 and 2.5. Naturally any specification of admissible ψ - for example the famous one defining q-calculus - has its own characteristic properties not pertaining to the standard case of Rota calculus realisation. Nevertheless the overall picture and system of statements depending only on GHW algebra is the same modulo some automatic replacements in formulas demonstrated in the sequel. The large part of that kind of job was already done in [2, 3].

The aim of this presentation is to give a general picture of the algebra of linear operators on polynomial algebra. The picture that emerges discloses the fact that any ψ-representation of finite operator calculus or equivalently - any ψ-representation of GHW algebra makes up an example of the algebraization of the analysis - naturally when constrained to the algebra of polynomials.

We shall delimit all our considerations to the algebra P of polynomials or sometimes to the algebra of formal series. Therefore the distinction in-between difference and differentiation operators disappears. All linear operators on P are both difference and differentiation operators if the degree of differentiation or difference operator is unlimited.

If all this is extended to Markowsky Q-umbral calculus then many of the results of ψ-calculus may be extended to Q-umbral calculus [11]. This is achieved under the almost automatic replacement of {D, x̂, id} generators of GHW or their ψ-representation {∂_{ψ}, x̂_{ψ}, id} by their Q-representation correspondents {Q, x̂_{Q}, id} - see definition 2.5.

The article is supplemented by the short indicatory glossaries of terms and notation used by Ward, Viskov, Markowsky, Roman on one side and the Rota-oriented notation on the other side.

2 Primary definitions, notation and general observations

In the following we shall consider the algebra P of polynomials P =F[x] over the field F of characteristic zero. All operators or functionals studied here are to be understood as linear operators on P. It shall be easy to see that they are always well defined.

Throughout the note while saying “polynomial sequence {p_{n}}_{o}^{∞}” we mean deg p_{n} = n; n ≥ 0 and we adopt also the convention that deg p_{n} < 0 iff p ≡ 0.

Consider ℑ - the family of functions‘ sequences (in conformity with Viskov notation ) such that: ℑ = {ψ; R ⊃ [a, b] ; q ∈ [a, b] ; ψ (q) : Z → F ; ψ_{0} (q) = 1 ; ψ_{n} (q) ≠ 0; ψ_{−n} (q) = 0; n ∈ N}. We shall call ψ = {ψ_{n} (q)}_{n≥0} ; ψ_{n} (q) ≠ 0; n ≥ 0 and ψ_{0} (q) = 1 an admissible sequence. Let now n_{ψ} denotes [2, 3]

n_{ψ} ≡ ψ_{n−1} (q) ψ_{n}^{−1} (q) .

Then

n_{ψ}! ≡ ψ_{n}^{−1} (q) ≡ n_{ψ} (n − 1)_{ψ} (n − 2)_{ψ} (n − 3)_{ψ} ....2_{ψ}1_{ψ}; 0_{ψ}! = 1

n\underline{k} ; ψ = n_{ψ} (n − 1)_{ψ} ... (n − k + 1)_{ψ} and ( \begin{matrix}n \\ k \end{matrix} )_{ψ} ≡ \frac{n\underline{k} ; ψ}{k_{ψ}!} and exp_{ψ}\{y\} = ∑_{k=0}^{∞} \frac{y^{k}}{k_{ψ}!}.

Definition 2.1. Let ψ be admissible. Let ∂_{ψ} be the linear operator lowering degree of polynomials by one defined according to ∂_{ψ}x^{n} = n_{ψ}x^{n−1} ; n ≥ 0. Then ∂_{ψ} is called the ψ-derivative.

Remark 2.1. The choice ψ_{n} (q)=[R (q^{n})!]^{−1} and R (x) = \frac{1−x}{1−q} results in the well known q-factorial n_{q}! = n_{q} (n − 1)_{q}!; 1_{q}! = 0_{q}! = 1 while the ψ-derivative ∂_{ψ} becomes now (n_{ψ} = n_{q}) the Jackson’s derivative [2, 3] ∂_{q}:

(∂_{q}φ) (x) = \frac{φ(x)−φ(qx)}{(1−q)x}.

Note also that that if ψ = {ψ_{n} (q)}_{n≥0} and φ = {φ_{n} (q)}_{n≥0} are two admissible sequences then [∂_{ψ} , ∂_{φ}]= 0 iff ψ = φ.

Definition 2.2. Let E^{y} (∂_{ψ}) ≡ exp_{ψ}\{y∂_{ψ}\} = ∑_{k=0}^{∞} \frac{y^{k}∂_{ψ}^{k}}{k_{ψ}!} . E^{y} (∂_{ψ}) is called the generalized transla-

tion operator.

Note 2.1. [2, 3]

E^{a} (∂_{ψ}) f(x) ≡ f(x +_{ψ} a) ; (x +_{ψ} a)^{n} ≡ E^{a} (∂_{ψ}) x^{n} ; E^{a} (∂_{ψ}) f = ∑_{n≥0} \frac{a^{n}}{n_{ψ}!}∂_{ψ}^{n}f;

and in general (x +_{ψ} a)^{n} ≠ (x +_{ψ} a)^{n−1}(x +_{ψ} a). Note also that in general (1 +_{ψ} (−1))^{2n+1} ≠ 0 ; n ≥ 0 though (1 +_{ψ} (−1))^{2n} = 0 ; n ≥ 1.

Note 2.2. [1] exp_{ψ} (x +_{ψ} y) ≡ exp_{ψ}{x} exp_{ψ}{y} - while in general exp_{ψ}{x + y} ≠ exp_{ψ}{x} exp_{ψ}{y}.

Possible consequent utilisation of the identity exp_{ψ} (x +_{ψ} y) ≡ exp_{ψ}{x} exp_{ψ}{y} is quite encouraging. It leads among others to “ψ-trigonometry” either ψ-elliptic or ψ-hyperbolic via introducing cos_{ψ}, sin_{ψ} [1], cosh_{ψ} , sinh_{ψ} or in general ψ-hyperbolic functions of m-th order {h_{j}^{(ψ)} (α)}_{j∈Z_{m}}defined according to [12]

R ∋ α → h_{j} (α) = \frac{1}{m} ∑_{k∈Z_{m}} ω^{−kj} exp_{ψ} (ω^{k}α) ; j ∈ Z_{m}, ω = exp (i\frac{2π}{m} ) .

where 1 < m ∈ N and Z_{m} = {0, 1, ..., m − 1}.

Definition 2.3. A polynomial sequence {p_{n}}_{o}^{∞} is of ψ-binomial type if it satisfies the recurrence

E^{y} (∂_{ψ}) p_{n} (x) ≡ p_{n} (x +_{ψ} y) ≡ ∑_{k≥0} ( \begin{matrix}n \\ k \end{matrix} )_{ψ}p_{k} (x) p_{n−k} (y) .

Polynomial sequences of ψ-binomial type [2, 3] are known to correspond in one-to-one manner to special generalized differential operators Q, namely to those Q = Q (∂_{ψ}) which are ∂_{ψ}-shift invariant operators [2, 3]. We shall deal in this note mostly with this special case i.e. with ψ-umbral calculus. However before to proceed let us supply a basic information referring to this general case of Q-umbral calculus.

Definition 2.4. Let P = F[x]. Let Q be a linear map Q : P → P such that: ∀_{p∈P} deg (Qp) = (deg p) − 1 (with the convention deg p = −1 means p = const = 0). Q is then called a generalized difference-tial operator [11] or Gel‘fond-Leontiev [7] operator.

Right from the above definitions we infer that the following holds.

Observation 2.1. Let Q be as in Definition 2.4. Let Qx^{n} = ∑_{k=1}^{n} b_{n,k}x^{n−k} where b_{n,1} ≠ 0 of

course. Without loose of generality take b_{1,1} = 1. Then ∃ {q_{k}}_{q≥2} ⊂ F and ∃ admissible ψ such that

Q = ∂_{ψ} + ∑_{k≥2} q_{k}∂_{ψ}^{k}  (2.2.1)

if and only if

b_{n,k} = ( \begin{matrix}n \\ k \end{matrix} )_{ψ} b_{k,k}; n ≥ k ≥ 1; b_{n,1} ≠ 0; b_{1,1} = 1.  (2.2.2)

If {q_{k}}_{q≥2} and an admissible ψ exist then these are unique.

Notation 2.1. In the case (2.2.2) is true we shall write : Q = Q (∂_{ψ}).

Remark 2.2. Note that operators of the (2.2.1) form constitute a group under superposition of formal power series (compare with the formula (S) in [13]). Of course not all generalized differencetial operators satisfy (2.2.1) i.e. are series just only in corresponding ψ-derivative ∂_{ψ} (see Proposition 3.1 ). For example [14] let Q = \frac{1}{2}Dx̂D − \frac{1}{3}D^{3}. Then Qx^{n} = \frac{1}{2}n^{2}x^{n−1} − \frac{1}{3}n\underline{3}x^{n−3} so according to Observation 2.1 n_{ψ} = \frac{1}{2}n^{2} and there exists no admissible ψ such that Q = Q (∂_{ψ}).

Observation 2.2. From theorem 3.1 in [11] we infer that generalized differential operators give rise to subalgebras ∑_{Q} of linear maps (plus zero map of course) commuting with a given generalized difference-tial operator Q. The intersection of two different algebras ∑_{Q_{1}} and ∑_{Q_{2}} is just zero map added.

The importance of the above Observation 2.2 as well as the definition below may be further fully appreciated in the context of the Theorem 2.1 and the Proposition 3.1 to come.

Definition 2.5. Let {p_{n}}_{n≥0} be the normal polynomial sequence [11] i.e. p_{0} (x) = 1 and p_{n} (0) = 0 ; n ≥ 1. Then we call it the ψ-basic sequence of the generalized difference-tial operator Q if in addition Q p_{n} = n_{ψ}p_{n−1}. Parallely we define a linear map x̂_{Q}: P → P such that x̂_{Q}p_{n} = \begin{matrix}( & n+1 & ) \\ (n+1) & _{ψ} \end{matrix} p_{n+1}; n ≥ 0. We call the operator x̂_{Q} the dual to Q operator.

When Q = Q (∂_{ψ}) = ∂_{ψ} we write for short: x̂_{Q(∂_{ψ})} ≡ x̂_{∂_{ψ}} ≡ x̂_{ψ} (see: Definition 2.9). Of course [Q, x̂_{Q}]= id therefore {Q, x̂_{Q}, id} provide us with a continuous family of generators of GHW in - as we call it - Q-representation of Graves-Heisenberg-Weyl algebra. In the following we shall restrict to special case of generalized differential operators Q, namely to those Q = Q (∂_{ψ}) which are ∂_{ψ}-shift invariant operators [2, 3] (see: Definition 2.6).

At first let us start with appropriate ψ-Leibnitz rules for corresponding ψ-derivatives. ψ-Leibnitz rules:

It is easy to see that the following hold for any formal series f and g: for ∂_{q}: ∂_{q} (f · g) = (∂_{q}f) · g + (Q̂f) · (∂_{q}g), where (Q̂f) (x) = f (qx);

for ∂_{R} = R (qQ̂) ∂_{o}: ∂_{R}(f • g)(z) = R (qQ̂) {(∂_{o}f)(z) • g(z) + f(0)(∂_{o}g)(z)}

where - note - R (qQ̂) x^{n−1} = n_{R}x^{n−1} ; (n_{ψ} = n_{R} = n_{R(q)} = R (q^{n})) and finally for ∂_{ψ} = n̂_{ψ}∂_{o}:

∂_{ψ}(f • g)(z) = n̂_{ψ}{(∂_{o}f)(z) • g(z) + f(0)(∂_{o}g)(z)}

where n̂_{ψ}x^{n−1} = n_{ψ}x^{n−1} ; n ≥ 1.

Example 2.1. Let Q (∂_{ψ}) = Dx̂D, where x̂f(x) = xf(x) and D = \frac{d}{dx}. Then ψ = \{[(n^{2})!]^{−1}\}_{n≥0}

and Q = ∂_{ψ}. Let Q (∂_{ψ}) R(qQ̂) ∂_{0} ≡ ∂_{R}. Then ψ = {[R (q^{n})!]^{−1}}_{n≥0} and Q = ∂_{ψ} ≡ ∂_{R}. Here R(z)

is any formal Laurent series; Q̂f(x) = f(qx) and n_{ψ} = R(q^{n}). ∂_{0} is q = 0 Jackson derivative which as a matter of fact - being a difference operator is the differential operator of infinite order at the same time:

∂_{o} = ∑_{n=1}^{∞} ( −1)^{n+1} \frac{x^{n−1}}{n!} \frac{d^{n}}{dx^{n}} .  (2.2.3)

Naturally with the choice ψ_{n} (q) = [R (q^{n})!]^{−1} and R (x) = \frac{1−x}{1−q} the ψ-derivative ∂_{ψ} becomes the Jackson’s derivative [2, 3] ∂_{q}:

(∂_{q}φ) (x) = \frac{1 − qQ̂}{(1 − q)}∂_{0}φ (x) .

The equivalent to (2.2.3) form of Bernoulli-Taylor expansion one may find [15] in Acta Eruditorum from November 1694 under the name “series univeralissima”.

(Taylor‘s expansion was presented in his “Methodus incrementorum directa et inversa” in 1715 - edited in London).

Definition 2.6. Let us denote by End(P) the algebra of all linear operators acting on the algebra P of polynomials. Let

∑_{ψ} = {T ∈ End(P); ∀ α ∈ F; [T; E^{α} (∂_{ψ})] = 0}.

Then ∑_{ψ} is a commutative subalgebra of End(P) of F -linear operators. We shall call these operators T : ∂_{ψ}-shift invariant operators.

We are now in a position to define further basic objects of “ψ-umbral calculus” [2, 3].

Definition 2.7. Let Q (∂_{ψ}) : P → P; the linear operator Q (∂_{ψ}) is a ∂_{ψ}-delta operator iff

1. Q (∂_{ψ}) is ∂_{ψ} - shift invariant;

2. Q (∂_{ψ}) (id) = const ≠ 0

The strictly related notion is that of the ∂_{ψ}-basic polynomial sequence:

Definition 2.8. Let Q (∂_{ψ}) : P → P; be the ∂_{ψ}-delta operator. A polynomial sequence {p_{n}}_{n≥0}; deg p_{n}= n such that:

1. p_{o} (x) = 1;

2. p_{n} (0) = 0; n > 0;

3. Q (∂_{ψ}) p_{n} = n_{ψ}p_{n−1} is called the ∂_{ψ}-basic polynomial sequence of the ∂_{ψ}-delta operatorQ (∂ψ).

Identification 2.1. It is easy to see that the following identification takes place: ∂_{ψ}-delta operator Q (∂_{ψ}) = ∂_{ψ}-shift invariant generalized differential operator Q. Of course not every generalized differential operator might be considered to be such.

Note: Let Φ (x; λ) = ∑_{n≥0} \frac{λ^{n}}{n_{ψ}!}p_{n} (x) denotes the ψ-exponential generating function of the ∂_{ψ}-

basic polynomial sequence {p_{n}}_{n≥0} of the ∂_{ψ}-delta operator Q ≡ Q (∂_{ψ}) and let Φ (0; λ) = 1. Then QΦ (x; λ) = λΦ (x; λ) and Φ is the unique solution of this eigenvalue problem. In view of Observation 2.2 we affirm that then exists such an admissible sequence φ that Φ (x; λ) = exp_{φ} [λx].

The notation and naming established by Definitions 2.7 and 2.8 serve the target to preserve and to broaden simplicity of Rota‘s finite operator calculus also in its extended “ψ-umbral calculus” case [2, 3]. As a matter of illustration of such notation efficiency let us quote after [2] the important Theorem 2.1 which might be proved using the fact that ∀ Q (∂_{ψ}) ∃! invertible S ∈ Σ_{ψ} such that Q (∂_{ψ}) = ∂_{ψ}S. ( For Theorem 2.1 see also Theorem 4.3. in [11], which holds for operators, introduced by the Definition 2.5).

Theorem 2.1. (ψ-Lagrange and ψ-Rodrigues formulas) Let {p_{n} (x)}_{n=0}^{∞} be ∂_{ψ}-basic polynomial sequence of the ∂_{ψ}-delta operator Q (∂_{ψ}). Let Q (∂_{ψ}) = ∂_{ψ}S_{∂_{ψ}}. Then for n > 0:

1. p_{n}(x) = Q (∂_{ψ})’ S_{∂_{ψ}}^{−n−1} x^{n} ;

2. p_{n}(x) = S_{∂_{ψ}}^{−n}x^{n} − \frac{n_{ψ}}{n} (S_{∂_{ψ}}^{−n} )’x^{n−1};

3. p_{n}(x) = \frac{n_{ψ}}{n} x̂_{ψ}S_{∂_{ψ}}^{−n}x^{n−1};

4. p_{n}(x) = \frac{n_{ψ}}{n} x̂_{ψ}(Q (∂_{ψ})’ )^{−1}p_{n−1}(x) (← Rodrigues ψ-formula ).

For the proof one uses typical properties of the Pincherle ψ-derivative defined bellow as well as x̂_{ψ} operator.

Definition 2.9. (compare with (17) in [7] ) The Pincherle ψ-derivative i.e. the linear map ’ : Σ_{ψ} → Σ_{ψ};

T ’ = T x̂_{ψ} - x̂_{ψ}T ≡[T_{∂_{ψ}} , x̂_{ψ}]

where the linear map x̂_{ψ} : P → P; is defined in the basis {x^{n}}_{n≥0} as follows

x̂_{ψ}x^{n} = \frac{ψ_{n+1} (q) (n + 1)}{ψ_{n} (q)}x^{n+1} = \begin{matrix}( & n + 1 & ) \\ (n + 1) & ψ \end{matrix} x^{n+1}; n ≥ 0

Observation 2.3. [2,3] The triples {∂_{ψ}, x̂_{ψ}, id} for any admissible ψ-constitute the set of generators of the ψ-labelled representations of Graves-Heisenberg-Weyl (GHW) algebra [16, 17, 18]. Namely, as easily seen [∂_{ψ}, x̂_{ψ}] = id. (compare with Definition 2.5)

Observation 2.4. In view of the Observation 2.3 the general Leibnitz rule in ψ-representation of Graves-Heisenberg-Weyl algebra may be written (compare with 2.2.2 Proposition in [17]) as follows

∂_{ψ}^{n} x̂_{ψ}^{m} = ∑_{k≥0} ( \begin{matrix}n \\ k \end{matrix} ) ( \begin{matrix}m \\ k \end{matrix} ) k! x̂_{ψ}^{m−k} ∂_{ψ}^{n−k}.  (2.2.4)

One derives the above ψ-Leibnitz rule from ψ-Heisenberg-Weyl exponential commutation rules exactly the same way as in {D, x̂, id} GHW representation - (compare with 2.2.1 Proposition in [17] ). ψ-Heisenberg-Weyl exponential commutation relations read:

exp{t∂_{ψ}}exp{ax̂_{ψ}} = exp{at}exp{ax̂_{ψ}}exp{t∂_{ψ}}.  (2.2.5)

To this end let us introduce a pertinent ψ-multiplication ∗_{ψ} of functions as specified below.

Notation 2.2.

x ∗_{ψ} x^{n} = x̂_{ψ}(x^{n}) = \begin{matrix}( & n+1 & ) \\ (n+1) & _{ψ} \end{matrix} x^{n+1}; n ≥ 0 hence x ∗_{ψ}1 = 1_{ψ} x ≢ x

x^{n}∗_{ψ} x = x̂_{ψ}^{n}( x ) = \frac{(n+1)!}{(n+1)_{ψ}!}x^{n+1}; n ≥ 0 hence 1∗_{ψ} x = 1_{ψ} x ≢ x therefore x ∗_{ψ}α1 = α1∗_{ψ} x = x ∗_{ψ}α = α∗_{ψ} x = α1_{ψ} x and ∀ x,α ∈ ; f(x)∗_{ψ} x^{n} = f(x̂_{ψ})x^{n}.

For k ≠ n x^{n}∗_{ψ} x^{k} ≠ x^{k}∗_{ψ} x^{n} as well as x^{n}∗_{ψ} x^{k} ≠ x^{n+k} - in general i.e. for arbitrary admissible ψ; compare this with (x +_{ψ}a)^{n} ≠ ( x +_{ψ}a)^{n−1}(x +_{ψ}a). In order to facilitate in the future formulation of observations accounted for on the basis of ψ-calculus representation of GHW algebra we shall use what follows.

Definition 2.10. With Notation 2.2 adopted let us define the ∗_{ψ} powers of x according to x^{n∗_{ψ}} ≡ x ∗_{ψ}x^{(n−1)∗_{ψ}} = x̂_{ψ}(x^{(n−1)∗_{ψ}}) = x ∗_{ψ} x ∗_{ψ} ... ∗_{ψ} x = \frac{n!}{n_{ψ}!}x^{n}; n ≥ 0.

Note that x^{n∗_{ψ}} ∗_{ψ} x^{k∗_{ψ}} = \frac{n!}{n_{ψ}!}x^{(n+k)∗_{ψ}} ≠ x^{k∗_{ψ}} ∗_{ψ} x^{n∗_{ψ}} = \frac{k!}{k_{ψ}!}x^{(n+k)∗_{ψ}} for k ≠ n and x^{0∗_{ψ}} = 1. This noncommutative ψ-product ∗_{ψ} is deviced so as to ensure the following observations:

Observation 2.5.

1. ∂_{ψ}x^{n∗_{ψ}} = nx^{(n−1)∗_{ψ}}; n ≥ 0

2. exp_{ψ}[αx] ≡exp {αx̂_{ψ}}1

3. exp [αx] ∗_{ψ} exp_{ψ}{βx̂_{ψ}}=exp_{ψ}{[α + β]x̂_{ψ}}

4. ∂_{ψ}(x^{k} ∗_{ψ} x^{n∗_{ψ}}) = (Dx^{k}) ∗_{ψ} x^{n∗_{ψ}} + x^{k} ∗_{ψ} (∂_{ψ}x^{n∗_{ψ}}) hence

5. ∂_{ψ}(f ∗_{ψ} g) = (Df) ∗_{ψ} g + f ∗_{ψ} (∂_{ψ}g) ; f, g - formal series 6. f(x̂_{ψ})g(x̂_{ψ}) 1 = f(x) ∗_{ψ} g̃(x) ; g̃(x) = g(x̂_{ψ})1.

Now the consequences of Leibnitz rule (e) for difference-ization of the product are easily feasible. For example the ψ-Poisson process distribution p_{m}(x);

∑_{m≥0} p_{m}(x) = 1;

p_{m}(x) = \frac{(λx)^{m}}{m!} ∗_{ψ} exp_{ψ}[−λx]  (2.2.6)

is the unique solution of its corresponding ∂_{ψ}-difference equation

∂_{ψ}p_{m}(x) + λp_{m}(x) = λp_{m−1}(x)m > 0 ; ∂_{ψ}p_{0}(x) = −λp_{0}(x)  (2.2.7)

As announced - the rules of ψ -product ∗_{ψ} are accounted for - as a matter of fact - on the basis of ψ-calculus representation of GHW algebra. Indeed; it is enough to consult Observation 2.5 and to introduce ψ-Pincherle derivation ˆ∂_{ψ} of series in powers of the symbol x̂_{ψ} as below. Then the correspondence between generic relative formulas turns out evident.

Observation 2.6. Let ˆ∂_{ψ} ≡ \frac{∂}{∂x̂_{ψ}} be defined according to ˆ∂_{ψ}f(x̂_{ψ}) = [∂_{ψ}, f(x̂_{ψ})]. Then ˆ∂_{ψ}x̂_{ψ}^{n} = nx̂_{ψ}^{n−1} ; n ≥ 0 and ˆ∂_{ψ}x̂_{ψ}^{n} 1 = ∂_{ψ}x^{n∗_{ψ}} hence [ˆ∂_{ψ}f(x̂_{ψ})]1 = ∂_{ψ}f(x) where f is a formal series in powers of x̂_{ψ} or equivalently in ∗_{ψ} powers of x.

As an example of application note how the solution of 2.2.7 is obtained from the obvious solution m(x̂_{ψ}) of the ˆ∂_{ψ}-Pincherle differential equation 2.2.8 formulated within G-H-W algebra generated by {∂_{ψ}, x̂_{ψ}, id}

ˆ∂_{ψ}_{m}(x̂_{ψ}) + λ_{m}(x̂_{ψ}) = λ_{m−1}(x̂_{ψ}) m > 0 ; ∂_{ψ}_{0}(x̂_{ψ}) = −λ_{0}(x̂_{ψ})  (2.2.8)

Namely : due to Observation 2.5 (f) p_{m}(x) = _{m}(x̂_{ψ})1, where

m(x̂_{ψ}) = \frac{(λx̂_{ψ})^{m}}{m!} exp_{ψ}[−λx̂_{ψ}].  (2.2.9)

3 The general picture

The general picture from the title above relates to the general picture of the algebra End(P) of operators on P as in the following we shall consider the algebra P of polynomials P = F[x] over the field F of characteristic zero.

We shall draw an over view picture of the situation distinguished by possibility to develop umbral calculus for any polynomial sequences {p_{n}}_{o}^{∞} instead of those of traditional binomial type only.

In 1901 it was proved [19] that every linear operator mapping P into P may be represented as infinite series in operators x̂ and D. In 1986 the authors of [20] supplied the explicit expression for such series in most general case of polynomials in one variable ( for many variables see: [21] ). Thus according to Proposition 1 from [20] one has:

Proposition 3.1. Let Q be a linear operator that reduces by one each polynomial. Let \{q_{n} \binom{}{x̂}\}_{n≥0} be an arbitrary sequence of polynomials in the operator x̂. Then T̂ = ∑_{n≥0} q_{n} \binom{}{x̂}Q^{n} defines a linear

operator that maps polynomials into polynomials. Conversely, if T̂ is linear operator that maps polynomials into polynomials then there exists a unique expansion of the form

T̂ = ∑_{n≥0} q_{n} \binom{}{x̂}Q^{n}.

It is also a rather matter of an easy exercise to prove the Proposition 2 from [20]:

Proposition 3.2. “ Let Q be a linear operator that reduces by one each polynomial. Let \{q_{n} \binom{}{x̂}\}_{n≥0}be an arbitrary sequence of polynomials in the operator x̂. Let a linear operator that maps polynomials into polynomials be given by

T̂ = ∑_{n≥0} q_{n} \binom{}{x̂}Q^{n}.

Let P (x; λ) = ∑_{n≥0} q_{n} (x)λ^{n} denotes indicator of T̂. Then there exists a unique formal series

Φ (x; λ); Φ (0; λ) = 1such that:

QΦ (x; λ) = λΦ (x; λ) .

Then also P (x; λ) = Φ (x; λ)^{−1} T̂Φ (x; λ).

Example 3.1. Note that ∂_{ψ}exp_{ψ}{λx} = λexp_{ψ}{λx}; exp_{ψ} [x] |_{x=0} = 1. (*) Hence for indicator of T̂; T̂ = ∑_{n≥0} q_{n} \binom{}{x̂}∂_{ψ}^{n} we have:

P (x; λ) = [exp_{ψ}{λx}]^{−1}T̂exp_{ψ}{λx}. (**)

After choosing ψ_{n} (q) = [n_{q}!]^{−1} we get exp_{ψ}{x} =exp_{q}{x}. In this connection note that exp_{o} (x) = \frac{1}{1−x} and exp(x) are mutual limit deformations for |x| < 1 due to:

\frac{exp_{o}(z)−1}{z} = exp_{o} (z) ⇒ exp_{o} (z) = \frac{1}{1−z} = ∑_{k=0}^{∞} z^{k}; |z| < 1 i.e.

exp (x) ⟵_{1←q} exp_{q} (x) = ∑_{n=0}^{∞} \frac{x^{n}}{n_{q}!} ⟶_{q→0} \frac{1}{1 − x}.

Therefore corresponding specifications of (*) such as exp_{o} (λx) = \frac{1}{1−λx} or exp(λx) lead to corresponding specifications of (**) for divided difference operator ∂_{0} and D operator including special cases from [20].

To be complete let us still introduce [2, 3] an important operator x̂_{Q(∂_{ψ})} dual to Q (∂_{ψ}).

Definition 3.1. (see Definition 2.5) Let {p_{n}}_{n≥0} be the ∂_{ψ}-basic polynomial sequence of the ∂_{ψ}-delta operator Q (∂_{ψ}). A linear map

x̂_{Q(∂_{ψ})}: P → P ; x̂_{Q(∂_{ψ})} = \begin{matrix}( & n+1 & ) \\ (n+1) & _{ψ} \end{matrix} p_{n+1}; n ≥ 0 is called the operator dual to Q (∂_{ψ}).

Comment 3.1. Dual in the above sense corresponds to adjoint in ψ-umbral calculus language of linear functionals’ umbral algebra (compare with Proposition 1.1.21 in [22] ).

It is now obvious that the following holds.

Proposition 3.3. Let \{q_{n} \binom{}{x̂_{Q(∂_{ψ})}}\}_{n≥0} be an arbitrary sequence of polynomials in the operator x̂_{Q(∂_{ψ})}. Then T = ∑_{n≥0} q_{n} \binom{}{x̂_{Q(∂_{ψ})}}Q (∂_{ψ})^{n} defines a linear operator that maps polynomials into

polynomials. Conversely, if T is linear operator that maps polynomials into polynomials then there exists a unique expansion of the form

T = ∑_{n≥0} q_{n} \binom{}{x̂_{Q(∂_{ψ})}}Q (∂_{ψ})^{n} .  (3.3.1)

Comment 3.2. The pair Q (∂_{ψ}) , x̂_{Q(∂_{ψ})} of dual operators is expected to play a role in the description of quantum-like processes apart from the q-case now vastly exploited [2, 3].

Naturally the Proposition 3.2 for Q (∂_{ψ}) and x̂_{Q(∂_{ψ})} dual operators is also valid. Summing up: we have the following picture for End(P) - the algebra of all linear operators acting on the algebra P of polynomials.

Q(P) ≡ ⋃_{Q} ∑_{Q} ⊂ End(P)

and of course Q(P) ≠ End(P) where the subfamily Q(P) (with zero map) breaks up into sum of subalgebras ∑_{Q} according to commutativity of these generalized difference-tial operators Q (see Definition 2.4 and Observation 2.2). Also to each subalgebra ∑_{ψ} i.e. to each Q (∂_{ψ}) operator there corresponds its dual operator x̂_{Q(∂_{ψ})}

x̂_{Q(∂_{ψ})} ∉ ∑_{ψ}

and both Q (∂_{ψ}) & x̂_{Q(∂_{ψ})} operators are sufficient to build up the whole algebra End(P) according to unique representation given by (3.3.1) including the ∂_{ψ} and x̂_{ψ} case. Summarising: for any admissible ψ we have the following general statement. General statement:

End(P) =[{∂_{ψ},x̂_{ψ}}] = [{Q (∂_{ψ}) , x̂_{Q(∂_{ψ})}}] = [{Q , x̂_{Q}}]

i.e. the algebra End(P) is generated by any dual pair {Q , x̂_{Q}} including any dual pair {Q (∂_{ψ}) , x̂_{Q(∂_{ψ})}} or specifically by {∂_{ψ},x̂_{ψ}} which in turn is determined by a choice of any admissible sequence ψ.

As a matter of fact and in another words: we have bijective correspondences between different commutation classes of ∂_{ψ}-shift invariant operators from End(P), different abelian subalgebras ∑_{ψ}, distinct ψ-representations of GHW algebra, different ψ-representations of the reduced incidence algebra R(L(S)) - isomorphic to the algebra Φ_{ψ} of ψ-exponential formal power series [2] and finally - distinct ψ-umbral calculi [7, 11, 14, 23, 2]. These bijective correspondences may be naturally extended to encompass also Q-umbral calculi, Q-representations of GHW algebra and abelian subalgebras ∑_{Q}. (Recall: R(L(S)) is the reduced incidence algebra of L(S) where L(S)={A; A⊂S; |A| < ∞}; S is countable and (L(S); ⊆) is partially ordered set ordered by inclusion [10, 2] ).

This is the way the Rota‘s devise has been carried into effect. The devise “much is the iteration of the few” [10] - much of the properties of literally all polynomial sequences - as well as GHW algebra representations - is the application of few basic principles of the ψ-umbral difference operator calculus [2]. ψ− Integration Remark : Recall : ∂_{o}x^{n} = x^{n−1}. ∂_{o} is identical with divided difference operator. ∂_{o} is identical with ∂_{ψ} for ψ = {ψ (q)_{n}}_{n≥0} ; ψ (q)_{n} = 1 ; n ≥ 0 . Let Q̂f(x)f(qx). Recall also that there corresponds to the “∂_{q} difference-ization” the q-integration [24, 25, 26] which is a right inverse operation to “q-difference-ization”. Namely

F (z) :≡ (∫_{q} φ) (z) := (1 − q) z ∑_{k=0}^{∞} φ (q^{k}z) q^{k}  (3.3.2)

i.e.

F (z) ≡ (∫_{q} φ) (z) = (1 − q) z (∑_{k=0}^{∞} q^{k}Q̂^{k}φ) (z) = ((1 − q) z \frac{1}{1 − qQ̂}φ) (z) .  (3.3.3)

Of course

∂_{q} ∘ ∫_{q} = id  (3.3.4)

as

\frac{1 − qQ̂}{(1 − q)}∂_{0} ((1 − q) ẑ\frac{1}{1 − qQ̂}) = id.  (3.3.5)

Naturally (3.3.5) might serve to define a right inverse operation to “q-difference-ization”

(∂_{q}φ) (x) = \frac{1−qQ̂}{(1−q)}∂_{0}φ (x)

and consequently the “q-integration “ as represented by (3.3.2) and (3.3.3). As it is well known the definite q-integral is an numerical approximation of the definite integral obtained in the q → 1 limit. Following the q-case example we introduce now an R-integration (consult Remark 2.1).

∫_{R} x^{n} = (x̂\frac{1}{R (qQ̂)}) x^{n} = \frac{1}{R (q^{n+1})}x^{n+1}; n ≥ 0  (3.3.6)

Of course ∂_{R} ∘ ∫_{R} = id as

R (qQ̂) ∂_{o} (x̂\frac{1}{R (qQ̂)}) = id.  (3.3.7)

Let us then finally introduce the analogous representation for ∂_{ψ} difference-ization

∂_{ψ} = n̂_{ψ}∂_{o}; n̂_{ψ}x^{n−1} = n_{ψ}x^{n−1}; n ≥ 1.  (3.3.8)

Then

∫_{ψ} x^{n} = (x̂\frac{1}{n̂_{ψ}} ) x^{n} = \frac{1}{(n + 1)_{ψ}} x^{n+1}; n ≥ 0  (3.3.9)

and of course

∂_{ψ} ∘ ∫_{ψ} = id  (3.3.10)

Closing Remark:

The picture that emerges discloses the fact that any ψ-representation of finite operator calculus or equivalently - any ψ-representation of GHW algebra makes up an example of the algebraization of the analysis - naturally when constrained to the algebra of polynomials. We did restricted all our considerations to the algebra P of polynomials. Therefore the distinction in-between difference and differentiation operators disappears. All linear operators on P are both difference and differentiation operators if the degree of differentiation or difference operator is unlimited. For example \frac{d}{dx} =

∑_{k≥1} \frac{d_{k}}{k!} Δ^{k} where d_{k} = [\frac{d}{dx}x\underline{k}]_{x=0} = ( −1)^{k−1} (k − 1)! or Δ = ∑_{n≥1} \frac{δ_{n}}{n!} \frac{d^{n}}{dx^{n}} where δ_{n} = [Δx^{n}]_{x=0} = 1.

Thus the difference and differential operators and equations are treated on the same footing.

An interesting task (which seems to be still ahead) is to investigate the Q representation of finite operator calculus as an example of the algebraization of the analysis - naturally when constrained to the algebra of polynomials.

4 Glossary

Here now come short indicatory glossaries of terms and notation used by Ward [1], Viskov [6, 7], Markowsky [11], Roman [27]- [31] on one side and the Rota-oriented notation on the other side.

Ward  Rota - oriented (this note)

[n] ; [n]!  n_{ψ}; n_{ψ}!
basic binomial coefficient [n, r] = \frac{[n]!}{[r]![n−r]!} ψ-binomial coefficient \binom{n}{k}_{ψ} ≡ \frac{n\underline{k} ; ψ}{k_{ψ}!}

Ward  Rota - oriented (this note)

D = D_{x} - the operator D  ∂_{ψ} - the ψ-derivative
D x^{n} = [n] x^{n−1}  ∂_{ψ} x^{n} = n_{ψ} x^{n−1}

(x + y)^{n}  (x +_{ψ} y)^{n}
(x + y)^{n} ≡ ∑_{r=0}^{n} [n, r] x^{n−r}y^{r}  (x +_{ψ} y)^{n} = ∑_{k=0}^{n} \binom{n}{k}_{ψ}x^{k}y^{n−k}

basic displacement symbol  generalized shift operator
E^{t}; t ∈Z  E^{y} (∂_{ψ}) ≡ exp_{ψ}{y∂_{ψ}}; y ∈F
Eφ(x) = φ(x + 1)  E(∂_{ψ})φ(x) = φ(x +_{ψ} 1)
E^{t}φ(x) = φ (x + \overline{t})  E^{y}(∂_{ψ})x^{n} ≡ (x +_{ψ} y)^{n}

basic difference operator  ψ-difference delta operator
Δ = E − id  Δ_{ψ} = E^{y}(∂_{ψ}) − id

Δ = ε(D) − id = ∑_{n=0}^{∞} \frac{D^{n}}{[n]!} − id

Roman  Rota - oriented (this note)

t; tx^{n} = nx^{n−1}  ∂_{ψ} - the ψ-derivative
∂_{ψ}x^{n} = n_{ψ}x^{n−1}
⟨t^{k}|p(x)⟩ = p^{(k)}(0)  [∂_{ψ}^{k}p(x)]|_{x=0}

evaluation functional  generalized shift operator
ϵ_{y}(t) = exp {yt}  E^{y}(∂_{ψ}) = exp_{ψ} {y∂_{ψ}}

⟨t^{k}|x^{n}⟩ = n!δ_{n,k}

⟨ϵ_{y}(t)|p(x)⟩ = p(y)  [E^{y}(∂_{ψ})p_{n}(x)]|_{x=0} = p_{n}(y)
ϵ_{y}(t)x^{n} = ∑_{k≥0} \binom{n}{k}x^{k}y^{n−k}  E^{y}(∂_{ψ})p_{n}(x) = ∑_{k≥0} \binom{n}{k}_{ψ}p_{k}(x)p_{n−k}(y)

Roman  Rota - oriented (this note)

formal derivative  Pincherle derivative
f′(t) ≡ \frac{d}{dt}f(t)  [Q(∂_{ψ})]‘≡ \frac{d}{d∂_{ψ}} Q(∂_{ψ})
\overline{f}(t) compositional inverse of  Q^{−1}(∂_{ψ}) compositional inverse of
formal power series f(t)  formal power series Q(∂_{ψ})

θ_{t}; θ_{t}x^{n} = x^{n+1}; n ≥ 0  x̂_{ψ}; x̂_{ψ}x^{n} = \frac{n+1}{(n+1)_{ψ}} x^{n+1}; n ≥ 0
θ_{t}t = x̂D  x̂_{ψ}∂_{ψ} = x̂D = N̂

∑_{k≥0} \frac{s_{k}(x)}{k_{ψ}!}t^{k} =  ∑_{k≥0} \frac{s_{k}(x)}{k_{ψ}!}z^{k} =

[g(\overline{f}(z))]^{−1} exp \{x\overline{f}(t)\}  s(q^{−1}(z)) exp_{ψ} \{xq^{−1}(z)\}
{s_{n}(x)}_{n≥0} - Sheffer sequence  q(t), s(t) indicators
for (g(t), f(t))  of Q(∂_{ψ}) and S_{∂_{ψ}}

g(t) s_{n}(x) = q_{n}(x) - sequence  s_{n}(x) = S_{∂_{ψ}}^{−1} q_{n}(x) - ∂_{ψ} - basic
associated for f(t)  sequence of Q(∂_{ψ})

The expansion theorem:  The First Expansion Theorem
h(t) = ∑_{k=0}^{∞} \frac{⟨h(t)|p_{k}(x)⟩}{k!}f(t)^{k}  T_{∂_{ψ}} = ∑_{n≥0} \frac{[T_{∂_{ψ}} p_{n}(z)]|_{z=0}}{n_{ψ}} Q(∂_{ψ})^{n}

p_{n}(x) - sequence associated for f(t) ∂_{ψ} - basic polynomial sequence {p_{n}}_{0}^{∞}

exp\{y\overline{f}(t)\} = ∑_{k=0}^{∞} \frac{p_{k}(y)}{k!}t^{k}  exp_{ψ}\{xQ^{−1}(x)\} = ∑_{k≥0} \frac{p_{k}(y)}{k!}z^{k}

The Sheffer Identity:  The Sheffer ψ-Binomial Theorem:
s_{n}(x + y) = ∑_{k=0}^{n} \binom{n}{k}p_{n}(y)s_{n−k}(x)  s_{n}(x +_{ψ} y) = ∑_{k≥0} \binom{n}{k}_{ψ}s_{k}(x)q_{n−k}(y)

Viskov  Rota - oriented (this note)

Viskov  Rota - oriented (this note)

θ_{ψ} - the ψ-derivative  ∂_{ψ} - the ψ-derivative
θ_{ψ} x^{n} = \frac{ψ_{n−1}}{ψ_{n}} x^{n−1}  ∂_{ψ} x^{n} = n_{ψ} x^{n−1}

A_{p} (p = {p_{n}}_{0}^{∞})  Q
A_{p} p_{n} = p_{n−1}  Q p_{n} = n_{ψ}p_{n−1}

B_{p} (p = {p_{n}}_{0}^{∞})  x̂_{Q}
B_{p} p_{n} = (n + 1) p_{n+1}  x̂_{Q} p_{n} = \frac{n+1}{(n+1)_{ψ}} p_{n+1}

E_{p}^{y} (p = {p_{n}}_{0}^{∞})  E^{y} (∂_{ψ}) = exp_{ψ}{y∂_{ψ}}
E_{p}^{y} p_{n}(x) = ∑_{k=0}^{n} p_{n−k}(x)p_{k}(y)  E^{y} (∂_{ψ}) p_{n}(x) =

= ∑_{k≥0} \binom{n}{k}_{ψ} p_{k}(x)p_{n−k}(y)

T − ε_{p}-operator:  E^{y} - shift operator:
T A_{p} = A_{p} T  E^{y}φ(x) = φ(x +_{ψ} y)

T - ∂_{ψ}-shift invariant operator:

∀_{y∈F} T E_{p}^{y} = E_{p}^{y}T  ∀_{α∈F} [T, E^{α}(∂_{ψ})] = 0

Q - δ_{ψ}-operator:  Q(∂_{ψ}) - ∂_{ψ}-delta-operator:
Q - ϵ_{p}-operator and  Q(∂_{ψ}) - ∂_{ψ}-shift-invariant and
Qx = const ≠ 0  Q(∂_{ψ})(id) = const ≠ 0

{p_{n}(x), n ≥ 0} - (Q, ψ)-basic  {p_{n}}_{n≥0} -∂_{ψ}-basic
polynomial sequence of the  polynomial sequence of the
δ_{ψ}-operator Q  ∂_{ψ}-delta-operator Q(∂_{ψ})

Viskov  Rota - oriented (this note)

ψ-binomiality property  ψ-binomiality property
Ψ_{y}s_{n}(x) =  E^{y}(∂_{ψ})p_{n}(x) =
= ∑_{m=0}^{n} \frac{ψ_{n}ψ_{n−m}}{ψ_{n}} s_{m}(x)p_{n−m}(y)  = ∑_{k≥0} \binom{n}{k}_{ψ}p_{k}(x)p_{n−k}(y)

T = ∑_{n≥0} ψ_{n}[V T p_{n}(x)]Q^{n}  T = ∑_{n≥0} \frac{[T p_{n}(z)]|_{z=0}}{n_{ψ}!} Q(∂_{ψ})^{n}

T Ψ_{y}p(x) =  T p(x +_{ψ} y) =
∑_{n≥0} ψ_{n}s_{n}(y)Q^{n}ST p(x)  ∑_{k≥0} \frac{s_{k}(y)}{k_{ψ}!}Q(∂_{ψ})^{k}S_{∂_{ψ}}T p(x)

Markowsky  Rota - oriented (this note)

L - the differential operator  Q
L p_{n} = p_{n−1}  Q p_{n} = n_{ψ}p_{n−1}

M  x̂_{Q}
M p_{n} = p_{n+1}  x̂_{Q} p_{n} = \frac{n+1}{(n+1)_{ψ}} p_{n+1}

L_{y}  F^{y} (Q) = ∑_{k≥0} \frac{p_{k}(y)}{k_{ψ}!}Q^{k}

L_{y} p_{n}(x) =  F^{y} (Q) p_{n}(x) =
= ∑_{k=0}^{n} \binom{n}{k}p_{k}(x)p_{n−k}(y)  = ∑_{k≥0} \binom{n}{k}_{ψ} p_{k}(x)p_{n−k}(y)

E^{a} - shift-operator:  E^{y} - ∂_{ψ}-shift operator:
E^{a} f(x) = f(x + a)  E^{y}φ(x) = φ(x +_{ψ} y)

G - shift-invariant operator: T - ∂_{ψ}-shift invariant operator:
EG = GE  ∀_{α∈F} [T, E^{α}(∂_{ψ})] = 0

Markowsky  Rota - oriented (this note)

G - delta-operator:  Q(∂_{ψ}) - ∂_{ψ}-delta-operator:
G - shift-invariant and  Q(∂_{ψ}) - ∂_{ψ}-shift-invariant and
Gx = const ≠ 0  Q(∂_{ψ})(id) = const ≠ 0

D_{L}(G)  G′ = [G(Q), x̂_{Q}]
L - Pincherle derivative of G  Q - Pincherle derivative
D_{L}(G) = [G, M]

{Q_{0}, Q_{1}, ...} - basic family  {p_{n}}_{n≥0} -ψ-basic
for differential operator L  polynomial sequence of the

generalized difference operator Q

binomiality property  Q - ψ-binomiali property
P_{n}(x + y) =  F^{y}(Q)p_{n}(x) =
= ∑_{i=0}^{n} \binom{n}{i}P_{i}(x)P_{n−1}(y)  = ∑_{k≥0} \binom{n}{k}_{ψ}p_{k}(x)p_{n−k}(y)

Acknowledgements

Discussions with Ewa Gra̧dzka are highly acknowledged. The author expresses his gratitude to her also for preparation of the LaTeX version of this contribution.

References

[1] M. Ward: Amer. J. Math. 58, 255 (1936).

[2] A. K. Kwaśniewski “Towards ψ-extension of Finite Operator Calculus of Rota” Rep. Białystok Univ. Inst. Comp. Sci. UwB/Preprint#05/July/2000; Rep. Math. Phys. in press

[3] A. K. Kwaśniewski “On extended finite operator calculus of Rota and quantum groups” Bia^{3}ystok Univ. Inst. Comp. Sci. UwB/Preprint#04/July/2000 Integral Transforms and Special Functions - in press

[4] R. P. Boas and Jr. R. C. Buck: Am. Math. Monthly 63, 626 (1959).

[5] R. P. Boas and Jr. R. C. Buck: Polynomial Expansions of Analytic Functions, Springer, Berlin 1964.

[6] O.V. Viskov: Soviet Math. Dokl. 16, 1521 (1975).

[7] O.V. Viskov: Soviet Math. Dokl. 19, 250 (1978).

[8] G.-C. Rota and R. Mullin: On the foundations of combinatorial theory, III. Theory of Binomial Enumeration in ”Graph Theory and Its Applications”, Academic Press, New York 1970.

[9] G. C. Rota, D.Kahaner and A. Odlyzko: J. Math. Anal. Appl. 42, 684 (1973).

[10] G. C. Rota: Finite Operator Calculus, Academic Press, New York 1975.

[11] G. Markowsky: J. Math. Anal. Appl. 63, 145 (1978).

[12] A. K. Kwaśniewski: Advances in Applied Clifford Algebras 9, 41 (1999).

[13] O.V. Viskov: Trudy Matiematicz‘eskovo Instituta AN SSSR 177, 21 (1986).

[14] A. Di Bucchianico and D.Loeb: J. Math. Anal. Appl. 92, 1 (1994).

[15] N. Ya. Sonin: Izw. Akad. Nauk 7, 337 (1997).

[16] C. Graves: Proc. Royal Irish Academy 6, 144 (1853-1857).

[17] P. Feinsilver and R. Schott: Algebraic Structures and Operator Calculus, Kluwer Academic Publishers, New York 1993.

[18] O.V. Viskov: Integral Transforms and Special Functions 1, 2 (1997).

[19] S. Pincherle and U. Amaldi: Le operazioni distributive e le loro applicazioni all‘analisi, N. Zanichelli, Bologna 1901.

[20] S. G. Kurbanov and V. M. Maximov: Dokl. Akad. Nauk Uz. SSSR 4, 8 (1986).

[21] A. Di Bucchianico and D.Loeb: Integral Transforms and Special Functions 4, 49 (1996).

[22] P. Kirschenhofer: Sitzunber. Abt. II Oster. Ackad. Wiss. Math. Naturw. Kl. 188, 263 (1979).

[23] A. Di Bucchianico and D.Loeb: J. Math. Anal. Appl. 199, 39 (1996).

[24] F. H. Jackson: Quart. J. Pure and Appl. Math. 41, 193 (1910).

[25] F. H. Jackson: Messenger of Math. 47, 57 (1917).

[26] F. H. Jackson: Quart. J. Math. 2, 1 (1951).

[27] S. M. Roman: J. Math. Anal. Appl. 87, 58 (1982).

[28] S. M. Roman: J. Math. Anal. Appl. 89, 290 (1982).

[29] S.M. Roman: J. Math. Anal. Appl. 95, 528 (1983).

[30] S. M. Roman: The umbral calculus, Academic Press, New York 1984.

[31] S. R. Roman: J. Math. Anal. Appl. 107, 222 (1985).

GEOMETRY OF LAGRANGIAN MANIFOLDS IN THERMODYNAMICS

V.P. MASLOV and A.S. MISCHENKO (Moscow)

Abstract

It is considered, that the classical thermodynamic properties of substance are defined by relations connecting volume, pressure, temperature, entropy and energy of the given substance. Generally substance is characterized by some number of magnitudes, which half is intensive, and half — by extensive magnitudes. From this point of view pressure and temperature are considered as intensive magnitudes, and volume and entropy - extensive magnitudes. The modern point of view consists that in a condition of a thermodynamic equilibrium the substance should be characterized by a point in the space R^{2n+1}(p, q, Φ), where coordinates q = (q_{1}, q_{2}, …, q_{n}) are intensive magnitudes, first two of which are pressure and temperature (q_{1} = P, q_{2} = −T), and coordinate p_{1}, p_{2}, …, p_{n}) are extensive coordinates first two from which are volume and entropy (p_{1} = V, p_{2} = S). First four coordinates (P, V, T, S) describe, so to tell, variables of a mechanical nature for homogeneous substances. generally follows to consider heterogeneous (i.e. multicomponent) systems, and also variables not mechanical nature (for example, electromagnetic properties). In any case, the space R^{2n+1}(p, q, Φ) is supplied by a contact structure, i.e. differential 1-form ω = dΦ − pdq, and the set of thermodynamic equilibrium states of substance is represented by a submanifold L ⊂ R^{2n+1}(p, q, Φ), such that ω = 0. Hence tha projection L_{0} ⊂ R^{2n}(p, q) is a Lagrangian submanifold in symplectic space R^{2n}(p, q) with the symplectic form Ω = dp∧dq. Function Φ is function of action on Lagrangian manifold L_{0}, dΦ = pdq. For classical thermodynamics it coincides with a thermodynamic potential (Φ = E + P V − T S).

By Gibbs ([1]) the energy E is a function of variables (V, S), as, however, and all remaining thermodynamic magnitudes. It hence, that Lagrangian manifold L_{0} bijectively is projected on a domain in the space R^{2}(P, S), i.e. the manifold L is defined by the graph of function E = E(V, S).

Implicitly Gibbs actually assumed, that the surface E = E(V, S), being noncompact, its any plane of support has by property, that touches a surface in each common point. This condition ensures realization of the following statement: from minimization thermodynamic potential at fixed P and T the positiveness of Hessian of function E = E(V, S), Hess_{(V,S)}E(V, S) > 0 follows. By Maslov ([3]) such condition are called essential. Then in essential condition are fulfilled local thermodynamic inequalities ([2]). Let’s consider function

Φ̃^{L}(q) = min_{q(x)=q;x∈L} Φ(x),

under condition of existence of the minimum in question. Consider a symplectic transformation φ of symplectic spaces

φ : R^{2n}(p, q)⟶R^{2n}(P, Q)

and Lagrangian manifold

Γ_{φ} ⊂ R_{4n}(P, p, Q, q), which is the graph of transformations φ. Let S - be function of action on Lagrangian manifold Γ_{φ}, dS = P dQ − pdq. Let’s assume, that manifold Γ_{φ} is uniquely projected on the space R^{2n}(Q, q). Then function S can be understood as function of variables (Q, q), S = S(Q, q). In this case the function S is called generating function of transformation φ.

contact

˜

be defined similar to manifold L_{1}.

Theorem 1. At an approaching choice of boundary conditions on manifold L and transformation φ the following formula takes place

Φ̃^{L_{1}}(Q) = min_{q} (S(Q, q) + Φ̃^{L}(q)) .

Similar, if φ_{1}, φ_{2}, …, φ_{n} is a sequence of symplectic transformations which admit generating functions

S_{1}(Q, q_{1}), S_{2}(q_{1}, q_{2}), …, S_{n}(q_{n−1}, q_{n}),

and L_{1} = φ̃_{1}φ̃_{2} ⋯ φ̃_{n}(L), then

Φ̃^{L_{1}}(Q) = min_{q_{1},q_{2},...,q_{n}} (S_{1}(Q, q_{1}) + S_{2}(q_{1}, q_{2}) + ⋯ + S_{n}(q_{n−1}, q_{n}) + Φ̃^{L}(q_{n})) .

The choice of boundary conditions should supply existence of a minimum at an evaluation of Φ̃^{L_{1}}. The theorem 1 supplies the map of insignificant points of manifold L into insignificant points of manifold L_{1} (compare [3]).

the Lagrangian L.

Theorem 2. Assume that the Lagrangian L(q, q̇) satisfies the conditions that the index of inertia of Hess_{q,q̇}L equals to (n, n). Let S(Q, q, t) be the generating function of transformation induced by Lagrangian L. Assume that there is a minimum

Φ̃^{L_{1}}(Q) = min_{q} (S(Q, q) + Φ̃^{L}(q)) .

Then

Hess_{Q}Φ̃^{L_{1}}(Q) < 0.

The theorem 2 ensures realization local thermodynamic inequalities in essential points of Lagrange manifold L_{1}.

Theorem 3. Let L be a Lagrange manifold which uniquely projected onto p–coordinates. Then at an approaching choice of boundary conditions in essential points the local thermodynamic inequalities are fulfilled, i.e.

Hess_{q}Φ^{L}(q) < 0.

As approaching boundary conditions for the theorem 3 the following condition can serve: Condition 1. The function E(p), dE = −qdp, E = Φ^{L}(p)−pq is locally convex upwards in all domain of definition G(p) ⊂ R^{n}(p) behind elimination of some compact set K ⊂ G(p).

The condition 1 is fulfilled for the majority of modelling examples of gases (ideal gas, Van der Waals gas , degenerated Fermi gas).

The theorems 1 and 2 allow to construct such Lagrangian manifolds, which are not projected uniquely on p - coordinate, but in all essential points satisfie to local thermodynamic inequalities..

[1] J.W.Gibbs, A method of geometrical representation of the thermodynamic properties of substatnces by mean of surfaces, Trans. Connect. Acad., 1873, II, Dec., p. 382-404

[2] L.D.Landau, E.M.Lifshitz, Statistical Physics, – ”Nauka”, Moscow, 1964

[3] V.P.Maslov, On a class of Lagrangian manifolds corresponding to variational problrms, problems of control theory an thermodynamics (in Russian)// Funct. Analysis and applications, Vol. 32, No 2, 1998

GEOMETRY OF GOURSAT FLAGS AND THEIR SINGULARITIES

OF CODIMENSION 2

PIOTR MORMUL

Institute of Mathematics, Warsaw University Banacha 2, 02-097 Warszawa, Poland e-mail: mormul@mimuw.edu.pl

April 9, 2001

1 Basic geometry – geometric classes – Jean’s strata

With every Goursat distribution – a particular rank–2 subbundle D in the tangent bundle to an n–dimensional manifold M (C^{∞} or analytic; n ≥ r + 2) such that the Lie square [D, D] of D is everywhere of rank 3, the Lie square of [D, D] is everywhere of rank 4, and so on until obtaining the full T M – associated is its flag of ascending induced subbundles D^{r} = D, D^{r−1} = [D, D], D^{r−2} = [[D, D], [D, D]], …, D^{0} = T M indexed by their coranks assumed to be constant independently of a point in M. The length of such a flag is r.

This is a very restrictive condition and G. germs are (excepting n = 4 and r = 2 – the classical situation of Engel, 1889) of codimension ∞ among germs of all distributions of fixed rank and corank. Yet, there exists an interesting trade off – the absence of functional parameters in local preliminary normal forms. In fact, Goursat distributions of corank r locally admit polynomial presentations of degrees ≤ r − 1 of Kumpera and Ruiz [KRu], using only real parameters, in numbers not exceeding r − 3, many of them possibly redundant. They also admit a trigonometric presentation springing from the kinematic model of a car pulling r − 1 passive trailers, developed by several authors in the 90s and refined to its limits by Jean [J] bringing in critical angles a_{1} = \frac{π}{2} , a_{j+1} = arctan(sin a_{j})). As a matter of fact, Kumpera and Ruiz discovered singularities hidden in flags of Goursat distributions. First attempts at defining them in a coordinate-free way were made in [BH] (p. 455), then in [CMPRe]. In [MonZ] singularities of Goursat flags were described in a canonical way, with a consistent use of the associated subflag of Cauchy–characteristic subdistributions.

Recalling, the basic singular features in the car’ presentation (attention: in that model, the last trailer has number 0, while trailer hooked to the car – number r−2, the car itself has number r − 1) are possible right angles between neighbouring trailers No k − 1 and k. They correspond to coalescences, at a point, of flag’ member D^{k+1} with the Cauchy–characteristic directions of two-step bigger member D^{k−1}. But flags exhibit also higher order singularities, implicitly present already in [J] (and constituting its strength), explicitly called tangent in [MonZ]. After [J], [CM] (last chap. 6), [MonZ] it is known that germs of G. flags of length r (or: distributions of corank r) can be stratified into F_{2r−3} (Fibonacci number) geometric classes encoded by words of length r over the alphabet {G, S, T} s. t. two first letters are always G and never a T goes directly after a G. A letter S is written in the code when a basic geometric coalescence takes place for the corresponding flag’ member D^{k}, members being indexed, we recall, backwards (or else: the right angle a_{1} occurs between two neighbouring trailers). A sequence STT… T inside a code corresponds to the next (i. e., closer to the car) neighbouring trailer making angle a_{2} with the second in the couple having the right angle, still next making angle a_{3} with the first next, and so as long as many T’s is in the sequence. This singular behaviour geometrically means that D^{k+1} is tangent at the reference point to the locus of the previous singularity ‘D^{k} in basic singular position’, plus D^{k+2} is tangent at that point to the locus of the singularity ‘ST’, plus D^{k+3} is tangent at that point to ‘STT’, and so on.

At the one next step the angle rule, or consecutive tangencies rule, breaks down and now the letters G are written in row until some next S (related to a new right angle in the configuration of trailers) appears.

Jean’s strata (the materializations of geometric classes on a given manifold carrying a G. flag) are, when non-empty, regular embedded submanifolds of codimensions that are easily computable. Namely, the codimension of a stratum having code C is equal to the number of letters S and T in C (cf. [M5], Sec. 1.4). The only codimension–0 stratum GGG… G, being open and dense, is non-empty on any n–dimensional manifold carrying a flag; G. germs at its points are equivalent to the classical chained model of von Weber (1898) – Cartan (1914)

– Goursat (1922) featuring no extra parameters:

dx^{2} − x^{3}dx^{1} = 0 ,
dx^{3} − x^{4}dx^{1} = 0 ,
dx^{4} − x^{5}dx^{1} = 0 ,
∗  ∗
dx^{r+1} − x^{r+2}dx^{k+1} = 0

(it should be understood as the germ at 0 ∈ ℝ^{n}(x^{1}, x^{2}, …, x^{n})).

The germs at points of the codimension–1 strata GG… GSG… G have been classified (for arbitrary length) in [M2], although only in [M3] the geometric description – based on [MonZ] – of the conditions securing the singular models was given. These singularities are simple in the singularity theory sense; parameters of K-R can be eliminated. The only invariant is the position 3 ≤ k ≤ r of the unique letter S in the r-letter code (the place in the flag where the unique coalescence of linear spaces at a point occurs). As a representative of the relevant orbit on n–dimensional manifolds can be taken the germ at 0 ∈ ℝ^{n}(x^{1}, x^{2}, …, x^{n}) of

dx^{2} − x^{3}dx^{1} = 0 ,
dx^{3} − x^{4}dx^{1} = 0 ,
∗  ∗
dx^{k} − x^{k+1}dx^{1} = 0 ,
dx^{1} − x^{k+2}dx^{k+1} = 0 ,
dx^{k+2} − (1 + x^{k+3})dx^{k+1} = 0 ,
dx^{k+3} − x^{k+4}dx^{k+1} = 0 ,
∗  ∗
dx^{r+1} − x^{r+2}dx^{k+1} = 0 .

In the present report we want to itemize what is known concerning the classification question for singularities of flags of codimension 2. That is – singularities at points in Jean’s strata having exactly two non–G letters in the code.

2 Strata having ST in their codes.

These strata have been recently classified in [M5] modulo one more singular (of codimension 3) feature of flags. In fact, for the whole strata GGSTG… G, G… GSTG, G… GST and for each GG… GSTG… G (at least three G’s in the beginning, at least two G’s in the end) less certain embedded submanifold of codimension 3 (not definable in the G, S, T language), we derive unique local models, the same in either of the categories C^{∞} or analytic. Again, only the position of the sequence ST in the code counts, and these singularities appear to be simple for Goursat flags of arbitrary length. When the letter S is at the k-th place in the code, 4 ≤ k ≤ r − 3, the only local model, on a manifold of dimension n, is the germ at 0 ∈ ℝ^{n}(x^{1}, x^{2}, …, x^{n}) of the rank–2 distribution described by the following r Pfaffian equations

dx^{2} − x^{3}dx^{1} = 0 ,
dx^{3} − x^{4}dx^{1} = 0 ,
∗  ∗
dx^{k} − x^{k+1}dx^{1} = 0 ,
dx^{1} − x^{k+2}dx^{k+1} = 0 ,
dx^{k+2} − x^{k+3}dx^{k+1} = 0 ,
dx^{k+3} − (1 + x^{k+4})dx^{k+1} = 0 ,
dx^{k+4} − (1 + x^{k+5})dx^{k+1} = 0 ,
dx^{k+5} − x^{k+6}dx^{k+1} = 0 ,
∗  ∗
dx^{r+1} − x^{r+2}dx^{k+1} = 0 .

The unique local model for the entire stratum GGSTGG… G is as the above for k = 3 except that its (k + 3)-th, i. e., sixth equation reads dx^{7} − x^{8}dx^{4} = 0 instead of dx^{7} − (1 + x^{8})dx^{4} = 0 . The models for G… GSTG and G… GST are the above for k = r − 2 and k = r − 1, respectively, with simplifications due to the fact that only the coordinates up to x^{r+2} enter the local description. This series of smooth, or analytic, local models consists of certain mentioned in the beginning (but highly specified in the course of a long proof) polynomial presentations of [KRu], of the G. germs in the respective geometric classes. After passing to the (dual) vector fields’ writing, polynomials are only of degree 2, because only one flag’s member is in a basic singular position, but there is plenty of constants in a preliminary presentation which should either be normalized (to 1 in the occurrence) or annihilated. Among the latter, certain are (much) more resistant. Surprisingly, the reason for that boils down to the arithmetical fact that – only for k ≥ 4 – there exist natural i’s such that 3k − 2 + i does not sit in the additive semigroup generated by 3 and 3k − 5. As one can easily check, these values of i are precisely 1, 4, …, 1 + 3(k − 4) (for inst., i = 1 and 4 for k = 5). They are just instances of the interesting distances put forward in [M2] (Def. 4.2 there for j = 1; note that that-time-k is now k − 3 ^{1}).

3 Strata having SS in their codes.

The work on these singularities is in progress. There are rather strong indications that the whole strata GGSSG… G are just single orbits of the local classification, with possible representatives

1 in Thm. 4.1 in [M2] the condition ‘(i − 1 ≥ k(j + 2) and i − 1 ∈ Z_{jk})’ should be replaced by ‘(i − 1 − k(j + 2) ∈ Z_{jk})’

dx^{2} − x^{3}dx^{1} = 0 ,
dx^{3} − x^{4}dx^{1} = 0 ,
dx^{1} − x^{5}dx^{4} = 0 ,
dx^{4} − x^{6}dx^{5} = 0 ,
dx^{6} − (1 + x^{7})dx^{5} = 0 ,
dx^{7} − x^{8}dx^{5} = 0 ,
dx^{8} − x^{9}dx^{5} = 0 ,
∗  ∗
dx^{r+1} − x^{r+2}dx^{5} = 0 .

Concerning the strata GG… GSSGG… G with at least three G’s in the beginning and at least three G’s in the end, when the letters S are at the k-th and (k + 1)-th places in the code, 4 ≤ k ≤ r − 4, we conjecture that, on an n–dimensional manifold, excepting certain embedded submanifold of codimension 3 (again, not definable in the G, S, T language) sitting in the stratum and cutting it into two disjoint parts, the germs at points of either part are equivalent to precisely one of the couple of germs at 0 ∈ ℝ^{n}(x^{1}, x^{2}, …, x^{n}) of

dx^{2} − x^{3}dx^{1} = 0 ,
dx^{3} − x^{4}dx^{1} = 0 ,
∗  ∗
dx^{k} − x^{k+1}dx^{1} = 0 ,
dx^{1} − x^{k+2}dx^{k+1} = 0 ,
dx^{k+1} − x^{k+3}dx^{k+2} = 0 ,
dx^{k+3} − (1 + x^{k+4})dx^{k+2} = 0 ,
dx^{k+4} − x^{k+5}dx^{k+2} = 0 ,
dx^{k+5} − (±1 + x^{k+6})dx^{k+2} = 0 ,
dx^{k+6} − x^{k+7}dx^{k+2} = 0 ,
∗  ∗
dx^{r+1} − x^{r+2}dx^{k+2} = 0 .

This should hold in both categories C^{∞} and analytic. As of now, this is, we repeat, a conjecture with work on it being in progress.

4 Strata having two not neighbouring S in their codes.

One should not, however, suppose that all codimension–2 singularities of Goursat flags are simple (admit only discrete local models). It is already not so in the geometric classes having the sequence SGS in the code, and at least three G’s in the beginning, as shown in [M4].

Before reviewing it in more detail, we want to note that first examples of continuous moduli in the Goursat world were found in geometric classes of codimension 3: GGGSTTGGG — [PRe], length 9, and, slightly later, GGSGSGSG — [M1], length 8.^{2} The latter example extends naturally and easily – see [M1], Rem. 4 – to the series of geometric classes GGSGSG… SG (when r is even) and GGGSGSG… SG (when r is odd) having precisely modality m = [\frac{r}{2}] − 3 wrt the classification of germs by diffeos acting in the base manifold: the orbits in these classes are exactly parametrized by m different real parameters. A far-reaching [but, it should be admitted, not yet sufficiently verified on various examples] conjecture of 1997 says that [\frac{r}{2}] − 3 is the maximal modality of germs of Goursat flags of length r.

2 Those findings were also briefly reported in [CMPRe].

Since the previous (2nd) Krynica Conference, an extensive work has been done on the geometric classes having the sequence SGS in their codes, and qualitatively new features, wrt the basic geometries ST and SS, revealed. In fact, excepting the class GGSGSGGG, a module of local C^{∞} or analytic classification was found not later than 3 steps after the behaviour SGS in the flag, see Thm. 1 in [M4]. Any germ in the class GG… GSGSGGG, with the first S being at the place k ≥ 4, appears equivalent to precisely one of the germs at 0 ∈ ℝ^{n}(x^{1}, …, x^{k+7}; x^{k+8}, …, x^{n}) of

dx^{2} − x^{3}dx^{1} = 0 ,
dx^{3} − x^{4}dx^{1} = 0 ,
∗ ∗
dx^{k} − x^{k+1}dx^{1} = 0 ,
dx^{1} − x^{k+2}dx^{k+1} = 0 ,
dx^{k+2} − (1 + x^{k+3}) dx^{k+1} = 0 ,
dx^{k+1} − x^{k+4}dx^{k+3} = 0 ,
dx^{k+4} − (1 + x^{k+5}) dx^{k+3} = 0 ,
dx^{k+5} − x^{k+6}dx^{k+3} = 0 ,
dx^{k+6} − (c + x^{k+7}) dx^{k+3} = 0

parametrized by c ∈ ℝ. This invariant parameter can be better exemplified by not reducing to 0 the constant in the one before last Pfaffian equation above. Quoting from Rem. 1 in [M4], in the family of KR pseudo-normal forms (for germs in the geometric class under consideration)

dx^{2} − x^{3}dx^{1} = 0 ,
dx^{3} − x^{4}dx^{1} = 0 ,
∗ ∗
dx^{k} − x^{k+1}dx^{1} = 0 ,
dx^{1} − x^{k+2}dx^{k+1} = 0 ,
dx^{k+2} − (1 + x^{k+3}) dx^{k+1} = 0 ,
dx^{k+1} − x^{k+4}dx^{k+3} = 0 ,
dx^{k+4} − (1 + x^{k+5}) dx^{k+3} = 0 ,
dx^{k+5} − (b + x^{k+6})dx^{k+3} = 0 ,
dx^{k+6} − (c + x^{k+7}) dx^{k+3} = 0 ,

the quantity c− 7b− \frac{5}{3}b^{2} is an invariant of the local smooth or analytic conjugacies preserving 0 ∈ ℝ^{n}. The analysis of the classes GG… GSGSG… G, but now with at least 4 letters G in the beginning and more letters G in the end, was being continued after closing [M4].

If the first S is at the place k ≥ 5, then (if the ambient dimension n is big enough) in the next group of 4 letters G there hides itself a new invariant independent of the one discussed above. Thus the modality of the SGS singularities is in many cases at least 2. One can say that the first module is concealed, provided k ≥ 4, in the group of three generic positions in the flag directly after the SGS behaviour, while the second module – provided k ≥ 5 – in the group of the following four generic positions. It is plausible that, when k is at least 6 and n is big enough, in certain next group of G’s a third module is located. And more, it is not even excluded that in geometric classes of unbounded length^{3} [GG… GSGS], with l letters G in the beginning, modality can be at least l − 2.

The geometric classes with sequences SG… GS in their codes have been little investigated yet. Nevertheless, the arguments pinpointing moduli in generic prolongations of the geometry SGS seem to guarantee the abundance of numeric invariants among germs featuring the geometries SG… GS, too.

References

[BH] R. Bryant, L. Hsu; Rigidity of integral curves of rank 2 distributions, Inventiones math. 114 (1993), 435 - 461.

[CM] [CM] M. Cheaito, P. Mormul; Rank–2 distributions satisfying the Goursat condition: all their local models in dimension 7 and 8, ESAIM: Control, Optimisation and Calculus of Variations ( URL: http://www.emath.fr/cocv/ ) 4 (1999), 137 – 158.

[CMPRe] [CMPRe] ——, ——, W. Pasillas-Lépine, W. Respondek; On local classification of Goursat structures, C. R. Acad. Sci. Paris 327 (1998), 503 – 508.

[J] [J] F. Jean; The car with n trailers: characterisation of the singular configurations, ESAIM: Control, Optimisation and Calculus of Variations ( URL: http://www.emath.fr/cocv/ ) 1 (1996), 241 – 266.

[KRu] [KRu] A. Kumpera, C. Ruiz; Sur l’équivalence locale des systèmes de Pfaff en drapeau, in: F. Gherardelli (ed.), Monge –Ampère Equations and Related Topics, Inst. Alta Math., Rome 1982, 201 – 248.

[MonZ] R. Montgomery, M. Zhitomirskii; Geometric approach to Goursat flags, preprint 1999, available at http://orca.ucsc.edu/˜ rmont.

[M1] P. Mormul; Local classification of rank–2 distributions satisfying the Goursat condition in dimension 9, in: P. Orro et F. Pelletier (eds.), Singularités et Géométrie sous-riemannienne, Travaux en cours 62, Hermann, Paris 2000,89 – 119.

[M2] ——; Goursat distributions with one singular hypersurface – constants important in their Kumpera-Ruiz pseudo-normal forms, preprint 185, Labo Topologie, Université de Dijon 1999.

[M3] ——; Goursat flags: classification of codimension–one singularities, J. Dyn. Control Syst. 6 (2000), 311 – 330.

3 a notion put forward in 1999 by Zhitomirskii

[M4] ——; Real moduli in codimension 2 in local classification of Goursat flags, preprint, Warsaw 2000.

[M5] P. Mormul; Simple codimension-two singularities of Goursat flags I: one flag’s member in singular position, preprint 01-01 (39) Inst. of Math., Warsaw University, available at http://www.mimuw.edu.pl/wydzial/raport mat.html

[PRe] W. Pasillas-Lépine, W. Respondek, On the geometry of Goursat structures, ESAIM: Control, Optimisation and Calculus of Variations ( URL: http://www.emath.fr/cocv/ ) 6 (2001), 119 – 181.

KV-COHOMOLOGY OF CONTACT MANIFOLDS

MICHEL NGUIFFO BOYOM

Abstract

Given a contact manifold (M, a) the group of a-preserving diffeomorphisms is denoted by G(a).We construct a Koszul-Vinberg chain complex C(a) on which G(a) acts by chaincomplex homomorphism. The G(a)-equivariant cohmology spaces of C(a) produce new contact invariants.

Michel NGUIFFO BOYOM Departement de Mathématiques Université Montpellier2, FRANCE e-mail: boyom@math.univ-montp2.fr

ON THE LEAVES OF A PREFOLIATION OF A K–DIFFERENTIAL SPACE

ANDRZEJ PIA̧TKOWSKI

March 7, 2001

Two years ago at the conference in Krynica, prof. W. Waliszewski has presented the definition of a 𝕂-differential space which is a generalization of the notions of a differentiable manifold and a differential space in the sense of Sikorski.

I would like to define the notion of a prefoliation of a 𝕂-differential space and to present theorems which describe some properties of leaves of the prefoliation.

First, I remind the definition of a 𝕂-differential space.

Let 𝕂 be an arbitrary field with a non-trivial norm (we can think here about ℝ or ℂ). Let M^{(0)} = {α : α : D_{α} → 𝕂} be a family of functions with an arbitrary family of sets {D_{α}} as domains. Define the set

.M^{(0)} = ⋃_{α∈M_{0}} D_{α}

which will be called the set of points of M^{(0)}. In the set of points of M^{(0)} define a topology topM^{(0)} as the weakest topology containing the family

{α^{−1}(B) : B is open in 𝕂 and α ∈ M^{(0)}}.

Next set

anM^{(0)} := {φ ∘ (α_{1}, ...α_{m}) : m ∈ ℕ and α_{1}, ..., α_{m} ∈ M^{(0)}

and φ is an analytical function defined on an open set in 𝕂^{m} with values in 𝕂}.

If A ⊂ .M^{(0)} then

M^{(0)}|A := {α|A ∩ D_{α} : α ∈ M^{(0)}}

and

M_{A}^{(0)} := {β : ∀_{p∈D_{β}}∃_{U∈topM^{(0)}}∃_{α∈M^{(0)}}(p ∈ U ∩ A ⊂ D_{β} ∧ U ⊂ D_{α} ∧ β|U ∩ A = α|U ∩ A)}.

It is easy to see that M^{(0)}|A ⊂ M_{A}^{(0)}.

Definition 1.1. The family M of functions with its values in 𝕂 is called a 𝕂-differential space, if the condition

anM = M = M_{.M}

is fulfilled.

Let M^{(0)} be an arbitrary family of functions with its values in 𝕂. One can prove the following

Proposition 1.2. The family M := (anM^{(0)})_{.M^{(0)}} is the smallest 𝕂-differential space with the set .M^{(0)} as the set of points, containing M^{(0)}.

Definition 1.3. The 𝕂-differential space M defined above is called the 𝕂-differential space generated by a family M^{(0)}.

Prof. W. Waliszewski has proved the following

Proposition 1.4. Let M be a 𝕂-differential space and A ⊂ .M. Then M_{A} ⊂ M if and only if A ∈ topM.

The above proposition gives

Corollary 1.5. Let M be a 𝕂-differential space and A ∈ topM. Then M|A = M_{A}.

Let M, N be 𝕂-differential spaces.

Definition 1.6. The mapping f : .M → .N is said to be smooth if for each β ∈ N we have β ∘ f ∈ M.

If the above condition holds then we write f : M → N.

It is obvious that if f : M → N then f : topM → topN i.e. f is a continuous mapping respective to the topologies topM and topN.

Let M be a 𝕂-differential space and p ∈ .M. Define M(p) := {α ∈ M : p ∈ D_{α}}.

Definition 1.7. Any 𝕂-linear mapping v : M(p) → 𝕂 such that

v(α · β) = v(α) · β(p) + α(p) · v(β)

for α, β ∈ M(p) is called a vector tangent to M at p. The family of all vectors tangent to M at p forms a vector space. This vector space is said to be tangent to M at p. It is denoted by T_{p}M.

It is easy to see that if v ∈ T_{p}M, U ∈ topM and α ∈ M then v(α) = v(α|U).

Let M, N be 𝕂-differential spaces and f : M → N. For each p ∈ .M, the mapping f determines a linear mapping (f_{∗})_{p} : T_{p}M → T_{f(p)}N called a tangent mapping. Namely, for v ∈ T_{p}M and β ∈ N(f(p)) we have

((f_{∗})_{p}v) (β) = v(β ∘ f).

Definition 1.8. The mapping f : M → N is called an immersion, if for each p ∈ .M the tangent mapping (f_{∗})_{p} is a monomorphism.

Now we define a prefoliation of a 𝕂-differential space. Let M = {α : α : D_{α} → 𝕂} be a 𝕂-differential space.

Definition 1.9. A pair (M, F) of 𝕂-differential spaces is called a prefoliation of M if

1) .F = .M,
2) topF is locally connected,
3) ∀_{p∈.M}∃_{U∈topF} (p ∈ U ∧ F_{U} = M_{U}).

Connected components of topF are called leaves of (M, F).

It is easy to see that the notion of a prefoliation is a generalization of the notion of the regular foliation, and of the Stefan foliation. Moreover, if (M, F) is a prefoliation of 𝕂differential space M then (topM, topF) is a topological foliation in the sense of Ehresmann.

It is not difficult to prove the following

Theorem 1.10. If (M, F) is a prefoliation of a 𝕂-differential space M then for the mapping f = id_{.M} we have f : F → M.

Proof. If β ∈ M then for each p ∈ .M denote by U_{p} such a neighbourhood of p respective to topF for which

F_{U_{p}} = M_{U_{p}}.

Thus we get an open covering {U_{p}}_{p∈.M} of the set .M respective to topF with

β|U_{p} ∈ M|U_{p} ⊂ M_{U_{p}} = F_{U_{p}}.

Therefore β ∈ F, since F is a 𝕂-differential space.

¿From the theorem we get

Corollary 1.11. Let (M, F) be a prefoliation of a 𝕂-differential space M. Then topM ⊂ topF and M ⊂ F.

We also have

Theorem 1.12. If (M, F) is a prefoliation of a 𝕂-differential space M then the mapping f = id_{.M} is an immersion.

Proof. Suppose that (f_{∗})_{p}(v) = 0 for some v ∈ T_{p}F. Then for each β ∈ M(p) we have v(β ∘ f) = 0, i.e. for each G ∈ topF with p ∈ G we have

v((β ∘ f)|G) = 0  (1.1.1)

Let α ∈ F (p). There exists U ∈ topF such that p ∈ U and F_{U} = M_{U} by the definition of a prefoliation. Therefore, α|D_{α} ∩ U ∈ F |U = F_{U} = M_{U} by Corollary 5. Thus there exist V ∈ topM and γ ∈ M such that p ∈ V ∩ U ⊂ D_{α} and V ⊂ D_{γ} and γ|V ∩ U = α|V ∩ U. Obviously γ ∈ M(p) and because of Corollary 11, V ∩ U ∈ topF. Consequently,

v(α) = v(α|V ∩ U) = v(γ|V ∩ U) = v(γ ∘ f|V ∩ U) = 0

by (1.1.1). Thus v = 0.

Corollary 1.13. If L is a leaf of a prefoliation (M, F) and φ : L ∋ q ↦ q ∈ .M then φ : F_{L} → M and φ is an immersion.

Corollary 1.14. If L is a leaf of a prefoliation (M, F) then M_{L} ⊂ F |L.

We show that even if id : F → M is an immersion and (topM, topF) is a topological foliation then (M, F) has not to be a prefoliation.

Example 1.15. Let M be the ℝ-differential space generated by the family of all continuous functions defined on ℝ with values in ℝ and let F be the ℝ-differential space generated by the family of all C^{∞} functions defined on ℝ with values in ℝ. Then (topM, topF) is the trivial topological foliation in the sense of Ehresmann of ℝ. Remark that f = id_{ℝ} : ℝ → ℝ is a smooth mapping F → M since M ⊂ F. One can prove that T_{p}F is a vector space of dimension 0 for each p ∈ ℝ. Therefore, (f_{∗})_{p} is a monomorphism for each p ∈ ℝ.

It is obvious that for each U ∈ topF = topM we have M_{U} = M|U ≠ F |U = F_{U} which means that (M, F) is not a prefoliation of an ℝ-differential space M.

Using the definition of a prefoliation, one can prove

Lemma 1.16. If (M, F) is a prefoliation of a 𝕂-differential space, then for each β ∈ F and for each p ∈ D_{β} there exist U ∈ topF and α ∈ M such that p ∈ U ⊂ D_{β} and β|U = α|U.

Definition 1.17. The leaf L of a prefoliation (M, F) is said to be proper if (topF)|L = (topM)|L.

We have the following

Theorem 1.18. Let (M, F) be a prefoliation of a 𝕂-differential space M and L be a proper leaf of (M, F). Then

M_{L} = F |L.

Proof. The inclusion M_{L} ⊂ F |L holds for each L by Corollary 14.

Let γ ∈ F |L. By a local connectedness of topF we have L ∈ topF and consequently γ ∈ F |L ⊂ F by Proposition 4 and Corollary 5. By Lemma 16, for each p ∈ D_{γ} there exists V ∈ topF such that p ∈ V ⊂ D_{γ} ⊂ L and there exists α ∈ M such that γ|V = α|V. Since L is proper, there exists W ∈ topM such that V = W ∩ L. Let p ∈ D_{γ} and define U := W ∩ D_{α} ∈ topM and \overline{α} := α|U ∈ M. Then

1) p ∈ U ∩ L ⊂ D_{γ} since U ∩ L ⊂ W ∩ L = V ⊂ D_{γ};

2) U ⊂ D\overline{α} (in fact, the equality holds);

3) \overline{α}|U ∩L = α|U ∩L = α|W ∩L∩D_{α} = γ|W ∩L∩D_{α} = γ|U ∩L since W ∩L∩D_{α} ⊂ V. By 1)-3) we have γ ∈ M_{L}.

Institute of Mathematics Technical University of Łódź aleja Politechniki 11, 90-924 Łódź POLAND e-mail: andpiat@ck-sg.p.lodz.pl

SUBMODULES OF VECTOR FIELDS AND ALGEBROIDS

PAUL POPESCU

Abstract

Algebroids and generalized algebroids defined in [3], particularly Courant algebroids considered in [1], define involutive and finite generated submodules of vector fields. But the most known algebroids are the Lie algebroids. They are considered by J. Pradines in [4] in connection with Lie groupoids, giving a coherent generalization of Lie theory. The third theorem of Lie related to the integration of Lie algebroids to Lie groupoids failed from the global viewpoint (Almeida-Molino, 1985, see [2, Theorem 4.4]). Most of people focused in particular to the global integration problem of Lie algebroids to Lie groupoids. Even the integration is not always possible, some particular integrable cases or obstructions regarding the integration were studied.

The algebroids considered in this paper are vector bundles which the module of sections fulfills the conditions of a Lie algebroid, except the Jacobi condition. But as our knowledge is, the relation between algebroids and vector fields, regarding the situation when a submodule of vector fields can be defined by the image of an algebroid, has not been yet studied. We prove that the necessary and sufficient condition for a submodule of vector fields to be defined by the image of an algebroid is that the module be involutive and finite generated. It means that the algebroid is a sufficient notion for these modules. As a first case when this situation occurs, we prove that a singular Riemannian foliation is always defined by an algebroid. As a second application, the canonical central anchored bundle of regular and singular Riemannian foliations is defined and the cojecture of P. Molino which asserts that the closures of leaves of a singular Riemannian foliation is also a singular Riemannian foliation is proved. In fact, Molino has left to prove only that the closures of leaves of a singular Riemannian foliation is a Stefan-Sussmann foliation [2, Chapter 6]; here we fill up this gap.

References

[1] Liu Z.-L., Weinstein A., Xu P., Manin triples for Lie algebroids, J. Diff. Geom., 45 (1997), 547-574, dg-ga/9508013.

[2] Molino P., Riemannian foliations, Progress in Math., vol.73 , Birkhauser, Boston, 1988.

[3] Popescu P., On Generalized Algebroids, New Developments in Differential Geometry, Budapest 1996, Kluwer Academic Publ., 1998, 329-342.

[4] Pradines J., Théorie de Lie pour les groupoides differentiables. Calcul différentiel dans la catégorie des groupoïdes infinitésimaux, C.R.Acad. Sci. Paris Sér. A Math. 264(1967), 245-248.

Paul POPESCU

Department of algebra and geometry, University of Craiova,

PO Box 4-66, Craiova, 1100, Romania

E-mail: paulpopescu@email.com

MODULAR CLASSES OF ANCHORED MODULES

PAUL POPESCU and MARCELA POPESCU

The categories of modules with differentials and the vector bundles with differentials, are defined by the first author in [5]. As explained in [4] or [5], they are two categories of vector bundles and two categories of modules; they are also two functors from the each category to the other. Some corresponding functors are induced from the categories of vector bundles with differentials to the corresponding categories of modules with differentials.

As it is proved in [7], there are two functors (called here the derived functors) from the two categories of anchored modules which allow linear connections to the two categories of Lie pseudoalgebras; in the presence of linear connections, these functors can be defined on all the categories of modules with differentials. A Lie pseudoalgebra (the derived Lie pseudoalgebra) can be associated with an anchored module which allows a linear connection. The derived Lie pseudoalgebra does not depend on the linear connection, but on the anchor, being an invariant object associated with an anchored module which allows a linear connection. Considering also the correspondences of morphisms, one define two pseudofunctors (i.e. a functor except sending the identity in the identity) and two natural functors (called the derived functors) respectively on the two categories of anchored modules which allow linear connections to the corresponding two categories of Lie pseudoalgebras.

We prove in the paper that the construction of the derived functors can be also performed using an other way, which is more suitable for vector bundles [6].

We show that a linear connection lifts on the derived module to a curvature free connection. The Picard groups related to anchored modules as well as the modular classes of preinfinitesimal modules and almost Lie structures are defined. The modular classes defined here agrees in a certain sense with the modular class of a Lie pseudoalgebra defined by J. Huebschmann [3].

References

[1] Greub W., Halperin S., Vanstone R., Connections, Curvature and Cohomology, Vol.I, Academic press, New York,1972.

[2] Higgins P.J., Mackenzie K., Algebraic construction in the category of Lie algebroids, J. Algebra 129 (1990), 194-230.

[3] Huebschmann J., Duality for Lie-Rine algebras and the modular class, J. Reine Angew. Math. 510 (1999), 103–159; dg-ga/9702008.

[4] Mackenzie K., Lie algebroids and Lie pseudoalgebras, Bull. London Math. Soc. 27 (1995), 97-147.

[5] Popescu P., Categories of modules with differentials, Journal of Algebra, 185 (1996), 50-73.

[6] Popescu P., Popescu M., Anchored Vector Bundles and Lie Algebroids, Lie algebroids and related topics in differential geometry, Banach Center, Warsaw, June 12-18 2000.

[7] Popescu P., The derived Lie pseudoalgebra of an anchored module (to appear).

Paul Popescu and Marcela Popescu

Department of Algebra and Geometry, University of Craiova PO Box 4-66, Craiova, 1100, Romania E-mail: paulpopescu@email.com, marcelapopescu@email.com

ERGODIC AND SPECTRAL PROPERITIES OF LAGRANGIAN AND

HAMILTONIAN DYNAMICAL SYSTEMS AND THEIR ADIABATIC PERTURBATIONS

ANATOLIY K. PRYKARPATSKY

Abstract

Any Lagrangian function on a closed finite-dimensional manifold M, when depending 2π–periodically on the evolution parameter generates so called Lagrangian flow. Its related group of diffeomorphisms on T (M) × 𝕊^{1} makes it possible to construct the set of normed (probabilistic) invariant measures on T (M) × 𝕊^{1}. The latter appears to be a convex set completely characterized by means of so called extreme points being at the same time due to a result of J. Mather ergodic measures of the Lagrangian flow under regard. On the other hand, there exists a natural mapping from the space of all invariant measures space mentioned above into the first homology group H_{1}(M; ℝ) of the manifold M via a well known Mather’s construction and some its generalizations subject to nonautonomous Hamiltonian flows on symplectic speces, whose image is exactly the measure homology of our Lagrangian or the corresponding Hamiltonian system. Its properties prove to be very important for detecting the corresponding ergodic measures, making use a new tool of its studying related with so called Legendrian transformations and Poincare -Cartan invariants. Moreover in the case when our Lagrangian function depends adiabatically on a small parameter ε↓ 0 through the expression εt ∈ ℝ/2πℤ, a suitable application of the Legendrian transformation together with the technique of Poincare -Cartan invariants makes it possible to investigate the existence and properties of so called adiabatic invariants and the corresponding limiting ergodic measures on T (M) × 𝕊^{1}. These same properties can be studied simultaneously making use also of the theory of spectral invariants applied to the generator of the corresponding Hamiltonian flow on the symplectic phase space T^{∗}(M).

Deptartment of Applied Mathematics at the AGH

al. Mickiewicza 30, 30-059 Kraków, POLAND

Department for Nonlinear Mathematical Analysis at the Academy of Sciences of Ukraine L’viv 79052, UKRAINE e-mail: prykanat@cybergal.com

INFINITE DIMENSIONAL LIE THEORY BY MEANS OF THE EVOLUTION MAPPING

TOMASZ RYBICKI

Abstract

An infinite dimensional Lie theory is said to be abstract if in the definition of the smooth structure on Lie groups charts are not required. Several abstract settings exist (Souriau, Chen, Omori), but usually they do not correspond to each other. The necessity of them is motivated by important examples and applications.

An infinite dimensional Lie group G with its Lie algebra 𝔤 is called regular if there is a bijective evolution mapping

evol_{G}^{r} : C^{∞}(ℝ, 𝔤) → ℭ^{∞}((ℝ, 0), (𝔊, 𝔢))

such that its evaluation at 1 ∈ ℝ is smooth. This notion has been introduced by Milnor. A concept generalizing regular Lie groups is proposed. In this concept the smooth structure is defined by means of the evolution mapping evol_{G}^{r}. Basic properties of Lie groups can be derived from our definition. New interpretations of the inheritance property, the third Lie theorem, and other facts are possible.

KEYWORDS: infinite dimensional Lie group, regularity

Department of Applied Mathematics at AGH

al. Mickiewicza 30, 30-059 Kraków

POLAND

tomasz@uci.agh.edu.pl

ON THE SET OF GEODESIC VECTORS OF A LEFT-INVARIANT METRIC

JÁNOS SZENTHE

If (M, <, >) is a Riemannian manifold its geodesic γ : R → M is said to be homogeneous if there is a 1-parameter group of isometries Φ : R × M → M of the Riemannian manifold such that

γ(τ) = Φ(τ, γ(0)), τ ∈ ℝ

holds. As the comprehensive paper of C. S. Gordon shows the existence of homogeneous geodesics in homogeneous Riemannian manifolds has essential geometric consequences [G]. Several results have been obtained recently concerning the existence of homogeneous geodesics. First it has been shown by V. V. Kajzer that if G is a Lie group and <, > a left-invariant Riemannian metric on G then the Riemannian manifold (G, <, >) has at least 1 homogeneous geodesic [Ka]. Generalizing this result of Kajzer it has been shown by O. Kowalski and J. Szenthe that if M = G/H is a homogeneous manifold and <, > an invariant metric on G/H then the homogeneous Riemannian manifold (G/H, <, >) has at least 1 homogeneous geodesic [Ko-Sz]. Moreover, it has been shown by Szenthe that if G is a compact semi-simple Lie group of rank ≥ 2 and <, > is a left-invariant Riemannian metric on G then the Riemannian manifold has infinitely many homogeneous geodesics [Sz]. In the study of

the set of homogenous geodesics of a homogeneous Riemannian manifold (G/H, <, >) the concept of geodesic vector proved to be convenient [Ko-V]. Let Φ : G × (G/H) → G/H be the canonical action, g the Lie algebra of G and Exp : g → G its exponential map. Put o = H ∈ G/H, fix a tangent vector v ∈ T_{o}(G/H) and consider the geodesic γ : R → G/H defined by v = γ̇(0). It is said that v is a geodesic vector if γ is a homogeneous geodesic of (G/H, <, >); in other words if

γ(τ) = Φ(Exp(τX), o), τ ∈ ℝ

holds with some X ∈ g. The study of the set of homogeneous geodesics of a homogenous Riemannian manifold is obviously reducible to the study of the set of its homogeneous vectors. However, it seems that the set of the geodesic vectors of a homogeneous Riemannian

manifold does not admit a simple description in general. Namely, O. Kowalski, S. Nikčević and Z. Vlašek in a joint paper have given several examples where the set of geodesic vectors have essentially different structure. In the lecture results are presented concerning the set of geodesic vectors of a homogeneous Riemannian manifold (G, <, >), where G is a compact semi-simple Lie group and <, > is a left-invariant Riemannian metric on G.

References

[ G] Gordon, C. S., Homogeneous Riemannian manifolds whose geodesics are orbits, Topics in Geometry in Memory of Joseph D’Atri, 1996, pp 155-174.

[ Ka] Kajzer, V. V., Conjugate points of left-invariant metrics on Lie groups, Sov. Math. 34 (1990), translation from Izv. Vyssh. Uchebn. Zaved. Mat. 342 (1990), pp 27-37.

[ Ko-N-V] Kowalski, O., Nikčević, S., Vlášek, Z., Homogeneous geodesics in homogenous Riemannian manifolds, to appear.

[ Ko-Sz] Kowalski, O., Szenthe, J., On the existence of homogeneous geodesics in homogeneous Riemannian manifolds, Geom. Dedicata 81 (2000), pp 209-214.

[ Ko-V] Kowalski, O., Vanhecke, L., Homogeneous Riemannian manifolds with homogeneous geodesics, Bull. Un. Mat. Ital., 5 (1991), pp 189-246.

[ Sz] Szenthe, J., Homogeneous geodesics of left-invariant metrics, Univ. Iagel. Acta Math. 38 (2000), pp 99-103.

Department of Geometry Eötvös Univ. Kecskeméti u. 10-12 Budapest, H-1053 HUNGARY e-mail: szenthe@ludens.elte.hu

ON THE STABILITY OF

SMOOTH DYNAMICAL SYSTEMS AND DIFFEOMORPHISMS

ANDRZEJ ZAJTZ

Abstract

General methods of studying global stability problems of smooth dynamical systems and diffeomorphisms are presented.

In particular, one proves that any complete C^{∞} vector field X in a Hilbert space E satisfying ⟨X(x), v⟩ ≥ δ > 0 for some constant field v and all x ∈ E (for instance, if ∥X − v∥ ≤ \frac{∥v∥}{2}) is globally rectifiable to v by a C^{∞} diffeomorphism of E. Thus any nonzero constant vector field in E is smoothly stable in a 0-order neighborhood.

Similarly one obtains the global stability (in a 1st-order neighborhood) of expansive non-resonant linear systems.

Institute of Mathematics, Pedagogical Academy ul. Podchora̧żych 2, 30-084 Kraków POLAND e-mail: smzajtz@cyf-kr.edu.pl