IV Conference

GEOMETRY AND TOPOLOGY OF MANIFOLDS

Krynica, 29 April – 4 May 2002

http://im0.p.lodz.pl/konferencje/krynica2002

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

Foreword 4

Organizers 5

List of participants 6

Titles of lectures 11

Abstracts 14

BLACKMORE, D. L., PRYKARPATSKY, Anatoliy, TANERI, Ufuk, Quantum holonomic computing via Lax type flows on Grassmannian manifolds and dual momentum mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

BOJARSKI, Bogdan, WEBER, Andrzej, Riemann–Hilbert problem: K-theory

and bordisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 BROWN, Ronald, Multiple groupoids: a non abelian tool for local–to–global problems 18 DESZCZ, Ryszard, GŁOGOWSKA Małgorzata, Curvature properities of some

hypersurfaces in spaces of constant curvature . . . . . . . . . . . . . . . . . 19 DOLIWA, Adam, The integrable geometric discretization of the Koenigs nets . . 20 DOMITRZ, Wojciech, On local classification of singular symplectic forms …. 21 EWERT–KRZEMIENIEWSKI, Stanisław, Notes on extended recurrent and

extended quasi-recurrent manifolds . . . . . . . . . . . . . . . . . . . . . . . 22 FEDORCHUK, V. M., FEDORCHUK, V. I., First–order differential invariants

of the splitting subgroup of the Poincaré group P (1, 4) . . . . . . . . . . . . 24 GRABOWSKI, Janusz, GRABOWSKA, Katarzyna, URBAŃSKI Paweł, Lie

bracket of affine bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 GER, Roman, The works of Professor Stanisaw Gołąb in the field of functional

equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 HAJDUK, Bogusław, Symplectic structures and diffeomorphisms . . . . . . . . . 30 HALL, Graham, Symmetries in 4-dimensional Lorentz manifolds . . . . . . . . . 31 HALL, Graham, Holonomy theory and 4–dimensional Lorentz manifolds . . . . . 38 HOTLOŚ, Marian, On hypersurfaces with type number two in space forms …. 39 KOCK, Anders, Combinatorics of the first neighbourhood of the diagonal …. 42 KORBAŠ, Július, On fibrations with Grassmann manifolds as fibers . . . . . . . 44 KUBARSKI, Jan, Tensor product of modules over non-unital algebras and

Lie-Rinehart algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 LECOMTE, Pierre, Spaces of differential operators as modules over the Lie algebra

of vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 MIKULSKI, Włodzimierz, Shortly on natural bundles . . . . . . . . . . . . . . . 48 MISHCHENKO, Alexandr, Poincaré duality of topological manifolds . . . . . . . 61 MORMUL, Piotr, Minimal nilpotent algebras in Goursat flags of lengths not

exceeding 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 MYKTYUK, Ihor, Invariant Kähler polarizations and geometry of vector fields . 69 NGUIFFO BOYOM, Michel, WOLAK, Robert, Local structures of KV-algebroids

and Lie algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 NIKITIN, Anatoly, Extended Poincaré parasupergroup with central charges … 71 OLSZAK, Zbigniew, On the holomorphic pseudosymmetry of Kähler manifolds . 72 PLACHTA, Leonid, Geometric aspects of invariants of finite type of knots in S^{3} 73 POPECSU, Marcela, POPECSU, Paul, On higher order geometry and induced

objects on subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 RYBICKI, Tomasz, A smooth structure on a locally compact topological group . . 76 SŁUKA, Karina, Properties of the curvature of Kähler–Norden manifolds …. 77

FOREWORD

Thank you for coming to Krynica! 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 2001 in Krynica (Poland) from 29.04.2002 to 4.05.2002.

The main purpose of the conference is to present an overview 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, Poisson 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,

• Committee on Mathematics of the Polish Academy of Sciences.

We hope that all of you will enjoy your stay in Krynica. We wish you success with your debates. Have a good time!

Organizers

ORGANIZERS

IV International Conference

GEOMETRY AND TOPOLOGY OF MANIFOLDS

is organized by

□ Institute of Mathematics of the Technical University of Łódź

Jan Kubarski [Chairman], (Łódź, Poland)

□ Institute of Mathematics of the Jagiellonian University, Cracow

Robert Wolak (Cracow, Poland)

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

Tomasz Rybicki (Cracow, Poland)

Scientific Committee

□ Dmitri ALEKSEEVSKY (Hull, UK) □ Ivan BELKO (Minsk, Belarus) □ Ronald BROWN (Bangor, UK) □ Stanisław BRZYCHCZY (Cracow, Poland) □ José F. CARIÑENA (Zaragoza, Spain) □ Rui Loja FERNANDES (Lisbon, Portugal) □ Janusz GRABOWSKI (Warsaw, Poland) □ Johannes HUEBSCHMANN (Lille, France) □ Július KORBAŠ (Bratislava, Slovakia) □ Oldřich KOWALSKI (Prague, Czech Republic) □ Jan KUBARSKI (Łódź, Poland) □ Alexandr MISHCHENKO (Moscow, Russia) □ Peter NAGY (Debrecen, Hungary) □ Anatoly NIKITIN (Kiev, Ukraine) □ Jean PRADINES (Toulouse, France) □ Tomasz RYBICKI (Cracow, Poland) □ János SZENTHE (Budapest, Hungary) □ Włodzimierz TULCZYJEW (Camerino, Italy) □ Robert WOLAK (Cracow, Poland)

LIST OF PARTICIPANTS

1. ALEKSEEVSKY, Dimitri, Department of Mathematics, University of Hull, Cottingham Road, HU6 7RX, Hull, UNITED KINGDOM, e-mail: D.V.Alekseevsky@Hull.ac.uk

2. BAJGUZ, Wiesław, Institute of Computer Science, University of Białystok, ul. Sosnowa 64, PL–15–887 Białystok, POLAND, e-mail: wbajguz@ii.uwb.edu.pl

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: niipulm@users.med.by

5. BENAMEUR, Moulay, Institut G. Desargues, Université Lyon 1, 43, Bd du 11-nov-1918, Bat. 101, 69622 Villeurbanne cedex, FRANCE, e-mail: benameur@desargues.univ–

lyon1.fr, http://www.desargues.univ-lyon1.fr/home/benameur/

6. 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/

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

8. BURSZTYN, Henrique, MSRI, Berkeley / ULB, Brussels, USA, e-mail: henrique@math.berkeley.edu

9. CA, Augusto, Guinea–Bissau Institute of Mathematics, 26 Rua Vitorino costa, Bissau, GUINEA–BISSAU, e-mail: gomes_foundation@hotmail.com

10. 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

11. CHIKE PRECIOUS AMAH, College of Continuing Education, Benin Study Centre, Upper Lawani Quarters, P. O. Box 14517, Benin City Edo State, NIGERIA

12. COLLETTI NEGREIROS, Caio José, University of Campinas (Unicamp), BRASIL, e-mail: caione@ime.unicamp.br

13. CORREIA, Patrico, Guinea–Bissau Institute of Mathematics, 26 Rua Vitorino costa, Bissau, GUINEA–BISSAU, e-mail: gomes_foundation@hotmail.com

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

15. DOLIWA, Adam, Institute of Theoretical Physics, Warsaw University, 00–681 Warszawa, ul. Hoża 69, POLAND, e-mail: doliwa@fuw.edu.pl, http://www.fuw.edu.pl/~doliwa

16. DOMITRZ, Wojciech, Faculty of Mathematics and Information Science, Warsaw University of Techology, plac Politechniki 1, 00–661 Warszawa, POLAND, e-mail: wdomitrz@ise.pw.edu.pl, domitrz@mini.pw.edu.pl 17. EHIGATOR TINO PRIASE, College of Continuing Education, Benin Study Centre,

Upper Lawani Quarters, P. O. Box 14517, Benin City Edo State, NIGERIA

18. EWERT–KRZEMIENIEWSKI, Stanisław, Institute of Mathematics, Technical University of Szczecin, al. Piastów 17, 70-310 Szczecin, POLAND, e-mail: ewert@arcadia.tuniv.szczecin.pl

19. FEDORCHUK, Vasyl, Institute of Mathematics, Pedagogical Academy, PL – 30–084 Kraków, ul. Podchora̧żych 2, Poland, 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

20. 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

21. GER, Roman, Institute of Mathematics, Silesian University, Bankowa 14, 40-007

Katowice, POLAND, http://www.imi.wsp.czest.pl/pracownicy/ger.html,

e-mail: romanger@us.edu.pl

22. GLANC, Barbara, Institute of Mathematics, Technical University of Szczecin, aleja Piastów 17, 70-310 Szczecin, POLAND, e-mail: glanc@arcadia.tuniv.szczecin.pl

23. GLAZEBROOK, James, Eastern Illinois University/University of Illinois at Urbana-Champaign, e-mail: cfjfg@eiu.edu, glazebro@math.uiuc.edu

24. GŁOGOWSKA, Małgorzata, Department of Mathematics, Agricultural University of Wrocław, ul. Grunwaldzka 53, 50–357 Wrocław, POLAND, e-mail: mglog@ozi.ar.wroc.pl

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

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

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

28. HOTLOŚ, Marian, Institute of Mathematics, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50–370 Wrocław, POLAND, e-mail: hotlos@im.pwr.wroc.pl

29. HUEBSCHMANN, Johannes, Université des Sciences et Technologies de Lille, UFR de Mathématiques, F–59 655 Villeneuve d’Ascq Cédex, FRANCE, e-mail: Johannes.Huebschmann@agat.univ-lille1.fr

30. KELECHI, Obi, Department of Mathematics, University of Lagos, Akoka, Cio Jide Joshua, 18 Okunola Street, Egbeda, Lagos State, NIGERIA, e-mail: batlimited@yahoo.com

31. KLEIN, Dariusz, Zakad Algebry, Pomorska Akademia Pedagogiczna, ul. Arciszewskiego 22 b, 76–200 Słupsk, POLAND, e-mail: klein@wsp.slupsk.pl

32. KOCK, Anders, Department of Mathematical Science, University of Aarhus, DK 8000 Aarhus C, DENMARK, e-mail: kock@imf.au.dk, http://home.imf.au.dk/kock/

33. KONDERAK, Jerzy, Dipartimento di Matematica, Università di Bari, Via Orabona 4, 70125 Bari, ITALY, e-mail: konderak@dm.uniba.it 34. KORBAŠ, Július, Faculty of Mathematics and Physics, Comenius University, Mlynská

dolina, SK–842 48 Bratislava 4, SLOVAKIA, e-mail: korbas@fmph.uniba.sk

35. 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/

36. KROT, Ewa, Institute of Computer Science, Białystok University, PL–15–887 Białystok, ul.Sosnowa 64, POLAND, e-mail: ewakrot@wp.pl

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

http://im0.p.lodz.pl/~kubarski

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

39. LECOMTE, Pierre, Departement of Mathematics of the University of Liege, Grande Traverse 12 (B37), B-4000 Liege (Sart Tilman), BELGIUM, e-mail: plecomte@ulg.ac.be,

http://www.forum.math.ulg.ac.be, http://www.ulg.ac.be/geothalg

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

41. LUCZYSZYN, Dorota, Institute of Mathematics, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50–370 Wrocław, POLAND, e-mail: luczysz@im.pwr.wroc.pl

42. MIKULSKI, Włodzimierz, Institute of Mathematics, Jagiellonian University, ul. W. Reymonta 4, 30–059 Kraków, POLAND, e-mail: mikulski@im.uj.edu.pl

43. MISHCHENKO, Alexandr, Moscow State University, Department of Mathematics, 119

992 GSP–2 Moscow, RUSSIA, http://higeom.math.msu.su/asmish,

e-mail: asmish@higeom.math.msu.su

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

45. MYKYTYUK, Ihor, Applied Mathematics Department, State University ”Lviv Polytechnica”, S. Bandera St. 12, 79013, Lvi’v, UKRAINE, e-mail: viva@iapmm.lviv.ua

46. NAGY, Péter Tibor, Debrecen University, Institute of Mathematics and Informatics, Department of Geometry, H-4010 Debrecen, P.O.B. 12, HUNGARY, e-mail: nagypeti@math.klte.hu

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

48. NIKITIN, Anatoly, Department of Applied Research, Institute of Mathematics of Nat. Acad. Sci. of Ukraine, 3 Tereshchenkivska St, 01601 Kiev–4, UKRAINE,

e-mail: nikitin@imath.kiev.ua, http://www.imath.kiev.ua/~nikitin/

49. NUNES DA COSTA, Joana Margarida, Departamento de Matemática, Universidade de Coimbra, Apartado 3008, 3001–454 Coimbra, PORTUGAL, e-mail: jmcosta@mat.uc.pt

50. NYONG, Godwin, GSA LIMITED, 12 Onasanya Street off Ishaga Road Surulere, Lagos State, NIGERIA, e-mail: gsalimitedng@hotmail.com 51. OLSZAK, Zbigniew, Institute of Mathematics, Wrocław University of Technology, Wybrzeże

Wyspiańskiego 27, 50–370 Wrocław, POLAND, e-mail: olszak@im.pwr.wroc.pl

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

53. PLACHTA, Leonid,Institute of Applied Problems of Mechanics and Mathematics of NAS of Ukraine, UKRAINE, e-mail: igor@iapmm.lviv.ua

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

55. PRADINES, Jean, Toulouse, FRANCE, e-mail: jpradines@wanadoo.fr, pradines@picard.ups-tlse.fr

56. PRYKARPATSKY, Anatolij K., Deptartment of Applied Mathematics at the AGH, al. Mickiewicza 30, bl. A4, 30–059 Kraków, POLAND, Deptartment of Nonlinear Math. Analysis at the Institute APMM of the Nat. Acad. of Sciences, Lviv, 79601 UKRAINE e-mail: prykanat@cybergal.com

57. 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

58. SAKALOS, Stefan, Faculty of Mathematics and Physics, Comenius University, Mlynská dolina, SK–842 48 Bratislava 4, SLOVAKIA, e-mail: sakalos@fmph.uniba.sk

59. SAWICZ, Katarzyna, Politechnika Częstochowska, Wydział Zarządzania, Instytut Ekonometrii i Informatyki, ul. Armii Krajowej 19B, 42–200 Częstochowa, POLAND, e-mail: k_sawicz@poczta.onet.pl

60. SCÁRDUA, Bruno, Institute of Mathematics, Federal University of Rio de Janeiro, Caixa Postal 68530, Rio de Janeiro – RJ, CEP. 21945-970, BRASIL, e-mail: scardua@im.ufrj.br

61. SŁUKA, Karina, Institute of Mathematics, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50–370 Wrocław, POLAND, e-mail: sluka@mazur.im.pwr.wroc.pl

62. SOBOLEWSKA, Kamila, Zakad Algebry, Pomorska Akademia Pedagogiczna, ul. Arc6iszewskiego 22 b, 76–200 Słupsk, POLAND, e-mail: Sobolewska@wsp.slupsk.pl

63. SPIRO, Andrea, Dipartimento di Matematica, Università di Camerino, Via Madonna delle Carceri, I-62032 Camerino (Macerata), ITALY, e-mail: andrea.spiro@unicam.it

64. STĘPIEŃ, Zofia, Institute of Mathematics, Technical University of Szczecin, al. Piastów 17, 70-310 Szczecin, POLAND, e-mail: stepien@arcadia.tuniv.szczecin.pl

65. SZENTHE, János, Department of Geometry, Eötvös University, 1518 Budapest, Pf. 120, HUNGARY, e-mail: szenthe@ludens.elte.hu

66. TOHAN CHRIS OKENYI, College of Continuing Education, Benin Study Centre, Upper Lawani Quarters, P. O. Box 14517, Benin City Edo State, NIGERIA

67. TOMAN, Henrietta, Debrecen University, Institute of Mathematics and Informatics, H-4010 Debrecen, P.O.B. 12, HUNGARY, e-mail: toman@math.klte.hu

68. TULCZYJEW, Włodzimierz, Camerino, ITALY, e-mail: tulczy@camserv.unicam.it 69. URBAŃSKI, Paweł, Faculty of Physics, University of Warsaw, Hoża 74, 00–682 Warszawa,

POLAND, e-mail: urbanski@fuw.edu.pl

70. VASSILIOU, Efstathios, Department of Mathematics, University of Athens, Panepistimiopolis, Athens 157 84, GREECE, e-mail: evassil@cc.uoa.gr

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

72. WALISZEWSKI, Włodzimierz, Faculty of Mathematics, University of Łódź, Banacha 22, 90–238 Łódź, POLAND, e-mail: faculty@math.uni.lodz.pl

73. WEBER, Andrzej, Institute of Mathematics, Warsaw University, ul. Banacha 2, 02–097 Warszawa, POLAND, e-mail: aweber@mimuw.edu.pl, http://www.mimuw.edu.pl/~aweber

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

75. 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

OPEN IN G PLEN ARY LECT URE

1. BROWN, Ronald  Multiple groupoids: a non abelian tool for

local-to-global problems

IN VIT ED PLEN ARY LECT URES

2. KORBAŠ, Július  On fibrations with Grassmann manifolds as fibers
3. MISHCHENKO, Alexandr  Poincaré duality of topological manifolds
4. PRYKARPATSKY, Anatolij Quantum holonomic computing via Lax type flows on

Grassmannian manifolds and dual momentum mappings

5. SZENTHE, János  Some isometric actions on Lorentz manifolds

IN VIT ED LECT URES

6. HALL, Graham  Holonomy theory and 4–dimensional Lorentz manifolds

7. NAGY, Péter Tibor  The classification of compact smooth Bol loops

8. TULCZYJEW, Włodzimierz Ehresmann jets and A. Weil’s point proches

SESSION — CELEBRATION OF the CENTENARY OF PROFESSOR GOŁĄB’s BIRTH

9. GER, Roman  The works of Professor Stanisaw Gołąb in the field

of functional equations

10. GOŁĄB–MEYER, Zofia  Master and disciple
11. MIKULSKI, Włodzimierz  Shortly on natural bundles
12. WALISZEWSKI, Włodzimierz On Gołąb’s pseudogroups
13. ZAJTZ, Andrzej  Professor Gołąb — life, work and our continuation

14. BAJGUZ, Wiesław  On embedding Peano continua in surfaces
15. BENAMEUR, Moulay  Homology of pseudodifferential operators

on foliations

16. BOJARSKI, Bogdan,  Riemann-Hilbert problem: K-theory and bordisms

WEBER, Andrzej

17. BROWN, Ronald  Knots and symbolic sculpture

(computer demonstration)

18. DESZCZ, Ryszard,  Curvature properities of some hypersurfaces
GŁOGOWSKA, Małgorzata  in spaces of constant curvature
19. DOLIWA, Adam  The integrable discretization of the Koenigs nets
20. DOMITRZ, Wojciech  On local classification of singular symplectic

forms

21. EWERT–KRZEMIENIEWSKI, Notes on extended recurrent and
Stanisław  extended quasi-recurrent manifolds
22. FEDORCHUK, V. M.  First-order differential invariants of the splitting
FEDORCHUK, V. I.  subgroup of the Poincaré group P (1, 4)

23. GRABOWSKI, Janusz,  Lie brackets of affine bundles

GRABOWSKA, Katarzyna,

URBAŃSKI, Paweł

24. HAJDUK, Bogusław  Symplectic structures and diffeomorphisms

25. HOTLOŚ, Marian  On hypersurfaces with type number two

in space forms

26. KOCK, Anders  Combinatorics of the first neighbourhood

of the diagonal

27. KONDERAK, Jerzy  Symplectic reduction theorem for manifolds

with indefinite metrics

28. KROT, Ewa  Q-difference Bernoulli-Taylor formula with

the rest term of the Cauchy type

29. KUBARSKI, Jan  Tensor product of modules over

non-unital algebras and Lie-Rinehart algebras

30. KWAŚNIEWSKI, Andrzej K. On an application of *psi-product and

GHW algebra

31. LECOMTE, Pierre  Spaces of differential operators as modules over

the Lie algebra of vector fields

32. LUCZYSZYN, Dorota  On the curvature of para-Kählerian manifolds
33. MISHCHENKO, Tatiana  Basic geometrical configurations

in the school geometry (in Russian)

34. MORMUL, Piotr  Minimal nilpotent algebras in Goursat flags of

lengths not exceeding 6

35. MYKYTYUK, Ihor  Invariant Kähler polarizations and geometry

of vector fields

36. NGUIFFO BOYOM, Michel Local structures of KV-algebroids and Lie algebroids

WOLAK, Robert

37. NIKITIN, Anatoly  Extended Poincaré parasupergroup with

central charges

38. NYONG, Godwin  Pure solution conveying geometrical posistion

and Euclidean space

39. OLSZAK, Zbigniew  On the holomorphic pseudosymmetry of

Kähler manifolds

40. PLACHTA, Leonid  Geomertic aspects of invariants of finite type

of knots in S^{3}

41. POPESCU, Marcela,  On higher order geometry and induced objects
POPESCU, Paul  on subspaces
42. RYBICKI, Tomasz  A smooth structure on a locally compact

topological group

43. SŁUKA, Karina  Properties of the curvature of Kähler–Norden manifolds
44. TOMAN, Henrietta  On the differentiability of n-loops

ABSTRACTS

QUANTUM HOLONOMIC COMPUTING VIA LAX TYPE FLOWS ON GRASSMANNIAN MANIFOLDS AND DUAL MOMENTUM MAPPINGS

D. L. BLACKMORE, ANATOLIJ PRYKARPATSKY, UFUK TANERI

Abstract

A general approach to holonomic quantum computing via Lax–type flows on Grassmann manifolds, based on the momentum mapping reduction and connection techniques is developed. It is shown that the associated holonomy groups can be effectively used in quantum computations of diverse practical problems.

Contents:

1. Introduction.

2. Symplectic structures on loop Grassmann manifolds.

3. An intrinsic loop Grassmannian structure and dual momentum mappings.

4. Quantum holonomy group.

References

[1] Wells R.O.: Differential analysis on complex manifolds, Prentice-Hall Inc., N.Y. 1973.

[2] Rajeev S. G., Kalyana R. S., Sidddhartha S.: Symplectic manifolds, coherent states and semiclassical approximation, J. Math. Phys. 35 (5) (1994), 2259–2269.

[3] Abraham R., Marsden J.: Foundation of mechanics, The Benjamin/Cummings Publ. Co, Masachusets, 1978.

[4] Bloch A. M.: Lax type flows on Grassmann manifolds, Contemporary Mathem. 68(1987), 39–50.

[5] Bloch A. M.: A completely integrable Hamiltonian systems associated with line fitting in complex vector space, Bull. AMS. 12(1985), 250–256.

[6] Faddeev L.D., Takhtadjan L. A.: Hamiltonian approach in the soliton theory, Springer, NY, 1986.

[7] Magri F.: On the geometry of soliton equations, Acta Applicandae Mathematicae, 41(1995), 247–270.

[8] Adams M. R., Harnard J., Hartubise J.: Dual momentum maps into loop algebras, Lett. Math. Phys., 20(1990).

[9] Rejman A. G., Semenov-Tian-Shansky M. A.: Reduction of Hamiltonian systems, affine Lie algebras and Lax equations, parts 1 &2, Inv. Mathem. 54(1979), 81– 100;63!(1981), 423–432.

[10] Prikarpatsky A. K., Mykytiuk I. V.: Algebraic aspects of the integrability of nonlinear dynamical systems on manifolds, Naukova Dumka, Kiev, 1991 (in Russian).

[11] Dittmann J., Rudolph G.: On a connection governing transport along (2+2)-density matrices, J. Geometry and Physics, 10 (1992), 93–106.

[12] Chruściński D., Jamiołkowski A.: Faza geometryczna: teoria i zastosowania, Nicolai Copernicus University Publisher, Torun Poland, 1996.

[13] Uhlmann A.: Parallel transport and ”quantum” holonomy along density operators, Rep. Math. Physics, 24(1986), 229–240.

[14] Olmo del M. A., Rodrigues M. A., Winternitz P.: Integrable systems based on SU(p,q) homogeneous manifold, Report CRM–1834, Sept. 1992, Université de Montreal QC, Canada.

[15] Sato M.: Soliton equations as dynamical systems on infinite - dimensional Grassmann manifolds, RIMS Kokyuroku, 439 (1981), 30–40.

[16] Prykarpatsky A. K., Blackmore D. L., Hentosh O. Y.: The finite dimensional Moser type reductions of modified Boussinesq and super Korteweg de Vries Hamiltonian systems via gradient holonomic algorithm and dual moment maps. New Frontiers in Physics, Proc. of Intern. Conf. at the Institute for Basic Res., Monteroduni, Italy, editor T. L. Gill, Hadronic Press, v.11 (1996), 271–292.

[17] Hazewinkel M.: Riccati and soliton equations, Report AMPR-9103, March 1995, CWI, Amsterdam, The Netherlands.

[18] Prykarpatsky Y.A., Blackmore D.L. and Samulyak R.: Ideals in Garssmann algebras and their integrability, J. of Nonl. Math. Phys., v.2, N., 1998.

[19] Fujii K., More on Optical Holonomic Quantum Computer, arXiv:quantph/0005129,v.1, 31 May 2000.

D. L. BLACKMORE Deptartment of Mathematical Sciences at the NJIT Newark NJ 07102 USA

Anatolij PRYKARPATSKY Deptartment of Applied Mathematics at the Academy of Mining and Metallurgy al. Mickiewicza 30, bl. A4, 30–059 Kraków POLAND, e-mail: prykanat@cybergal.com and Deptartment of Nonlinear Math. Analysis at the Institute APMM of the Nat. Acad. of Sciences Lviv,79601 UKRAINE

Ufuk TANERI Deptartment of Mathematics at the Eastern Mediterranean University Famagusta, via Mersin 10 TURKEY

RIEMANN–HILBERT PROBLEM: K–THEORY AND BORDISMS^{1}

BOGDAN BOJARSKI and ANDRZEJ WEBER

Abstract

We present classical and generalized Riemann–Hilbert problem in several contexts arising from K–theory and bordism theory. The language of Fredholm pairs turns out to be useful and unavoidable. We propose an abstract formulation of a notion of bordism in the context of Hilbert spaces equipped with splittings.

Bogdan BOJARSKI Institute of Mathematics Polish Academy of Sciences ul. Śniadeckich 8, P.O.Box 137 00-950 Warszawa POLAND e-mail: bojarski@impan.gov.pl

Andrzej WEBER Institute of Mathematics Warsaw University ul. Banacha 2, 02–097 Warszawa POLAND, e-mail: aweber@mimuw.edu.pl

http://www.mimuw.edu.pl/~aweber

1http://www.mimuw.edu.pl/~aweber/ps/bowe4.ps

MULTIPLE GRUPOIDS:

A NON ABELIAN TOOL FOR LOCAL–TO–GLOBAL PROBLEMS

RONALD BROWN

Abstract

A 1–fold groupoid is just a groupoid and and n–fold groupoid is a set with n groupoid structures any two of which satisfy an interchange law. The aim of the talk is to show the intuitions behind how types of n–fold groupoid have been used in homotopy theory and the cohomology of groups to give new calculations, with a view to encouraging their use in differential situations.

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/

CURVATURE PROPERTIES

OF SOME HYPERSURFACES IN SPACES OF CONSTANT CURVATURE

RYSZARD DESZCZ and MAŁGORZATA GŁOGOWSKA

Abstract

We present curvature properties of pseudosymmetry type of some hypersurfaces in semi-Riemannian spaces of constant curvature. Results related to this talk are contained in the following papers: [1], [2], [3] and [4].

References

[1] R. Deszcz and M. Głogowska, Some examples of nonsemisymmetric Riccisemisymmetric hypersurfaces, Colloq. Math., in print.

[2] R. Deszcz and M. Głogowska, On nonsemisymmetric Ricci-semisymmetric warped product hypersurfaces, Dept. Math., Agricultural Univ. Wrocław, Report No. 90, 2000.

[3] R. Deszcz, M. Głogowska, M. Hotloś, and L. Verstraelen, On some generalized Einstein metric conditions on hypersurfaces in semi-Riemannian space forms, Dept. Math., Agricultural Univ. Wrocław, Report No. 99, 2001.

[4] M. Głogowska, On a curvature characterization of Ricci-pseudosymmetric hypersurfaces, Dept. Math., Agricultural Univ. Wrocław, Report No. 97, 2001.

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

Małgorzata GŁOGOWSKA Department of Mathematics, Agricultural University of Wrocław ul. Grunwaldzka 53 PL - 50-357 Wrocław POLAND e-mail: mglog@ozi.ar.wroc.pl

THE INTEGRABLE GEOMETRIC DISCRETIZATION OF THE KOENIGS NETS

ADAM DOLIWA

Abstract

We introduce the Koenigs lattice, which is new integrable reduction of the quadrilateral lattice (discrete conjugate net). The discretization is performed by the natural extension of the basic geometric properties of the Koenigs net to the discrete level. We construct also the Darboux-type transformation of the Koenigs lattice and we show permutability of superpositions of such transformations, thus proving integrability of the Koenigs lattice. Details can be found in nlin.SI/0203011.

Institute of Theoretical Physics Warsaw University 00–681 Warszawa, ul. Hoża 69 POLAND e-mail: doliwa@fuw.edu.pl

http://www.fuw.edu.pl/~doliwa

ON LOCAL CLASSIFICATION OF SINGULAR SYMPLECTIC FORMS

WOJCIECH DOMITRZ

Abstract

Let ω be a closed 2-form on ℝ^{2n}, ω is called the singular form if ω^{n} = fΩ, where Ω is the volume form and f is a smooth function f : ℝ^{2n} → ℝ such that Σ_{2} = f^{−1}(0) ≠ ∅. In [3], [5] and [2] it was shown that there are five generic types of singularities for germs of 2-forms on ℝ^{4}. All these generic types, among which, one is unstable and has moduli parameters, are determined by the geometry of the pullback i^{∗}ω to the hypersurface of degeneration i : Σ_{2} → ℝ^{4}.

If germs of singular symplectic forms are equivalent then their pullbacks to the hypersurfaces of degeneration are equivalent. We consider the following problem :

Does the pullback of the singular symplectic form to its hypersurface of degeneration determine the equivalence class of the form?

References

[1] W. Domitrz, S. Janeczko and Z. Pasternak-Winiarski Geometry and representation of the singular symplectic forms, preprint.

[2] M. Golubitsky and D. Tischler, An example of moduli for singular symplectic forms, Inventiones Math., 38 (1977), 219-225.

[3] J. Martinet, Sur les singularités des formes differentielles, Ann. Inst. Fourier (Grenoble), 20 (1970), 95-178.

[4] J. Moser, On volume elements on manifold, Trans. Amer. Math. Soc., 120 (1965), 280-296.

[5] R. Roussarie, Modèles locaux de champs et de formes, Astérisque, 30, (1975), 1-181.

Faculty of Mathematics and Information Science

Warsaw University of Techology

plac Politechniki 1, 00–661 Warszawa

POLAND

e-mail: wdomitrz@ise.pw.edu.pl, domitrz@mini.pw.edu.pl

NOTES ON EXTENDED RECURRENT AND EXTENDED QUASI-RECURRENT MANIFOLDS

STANISŁAW EWERT–KRZEMIENIEWSKI

Abstract

Recently M. Prvanović ([P] ) introduced a type of manifold (M, g) whose (0, 4) curvarture tensor R satisfies

∇_{Z}R(X, Y, U, V) = ; \begin{matrix}A(Z) [R(X, Y, U, V) + (β − ψ) G(X, Y, U, V)] + \\ \frac{β}{2} & [A(X)G(Z, Y, U, V) + A(Y)G(X, Z, U, V)+ \\ A(U)G(X, Y, Z, V) + A(V)G(X, Y, U, Z)] , \end{matrix}  (1)

where β, ψ are functions on M, A is a closed form satisfying βA(\frac{∂}{∂x^{r}}) = \frac{∂ψ}{∂x^{r}} and G(X, Y, U, V) = g(Y, U)g(X, V) − g(Y, V)g(X, U). She proved that in a neighbourhood of a generic point the associated 1-form A is concircular, i. e.

(∇A) (X, Y) = F g(X, Y) + HA(X)A(Y)

holds for some function F, H, and found the local form of the metric ( [P]). The condition (1) can be considered as a generalisation of the well known notion of a recurrent manifold (∇_{Z}R = a(Z)R, ([W])) as well as of generalized recurrent manifold introduced by Dubey ([D])

∇_{Z}R(X, Y, U, V) = K(Z)R(X, Y, U, V) + L(Z)G(X, Y, U, V).  (2)

On the other hand Chaki ([Ch]) introduced and studied a type of manifolds satisfying

∇_{Z}R(X, Y, U, V) = ; \begin{matrix}2a(Z)R(X, Y, U, V) + a(X)R(Z, Y, U, V)+ \\ a(Y)R(X, Z, U, V) + a(U)R(X, Y, Z, V) + a(V)R(X, Y, U, Z) \end{matrix}

known as a pseudo-symmetric ( in the sense of Chaki) or quasi-recurrent.

We discuss the condition including the above ones.This can be reduce to the form

∇_{Z}R(X, Y, U, V) = ; \begin{matrix}2a(Z)R(X, Y, U, V) + a(X)R(Z, Y, U, V)+ \\ a(Y)R(X, Z, U, V) + a(U)R(X, Y, Z, V) + a(V)R(X, Y, U, Z) \\ 2b(Z)G(X, Y, U, V) + b(X)G(Z, Y, U, V)+ \\ b(Y)G(X, Z, U, V) + b(U)G(X, Y, Z, V) + b(V)G(X, Y, U, Z). \end{matrix}  (3)

We derive the formulas for the Ricci tensor, its covariant derivative and the Riemann curvature tensor. It appears that the manifold is of almost constant curvature. Making use of tensor charcterisation of subprojective spaces ([K]) we prove

Theorem 1. Let M be a conformally flat manifold, dimM > 4, whose curvature satisfy (3). If the curvature tensor R and the 1-form a do not vanish on any dense subset of M, then M is subprojective manifold.

This enables finding the local form of the metric of manifolds under consideration.

References

[Ch] M. C. Chaki, On pseudo symmetric spaces, Acta Math. Hungar. 54 (1989), no. 3-4, 185–190.

[D] R. S. D. Dubey, Generalized recurrent spaces, Indian J. Pure Appl. Math.10 (1979), no.12, 1508–1513.

[E-K 1] S. Ewert-Krzemieniewski, On some generalisation of recurrent manifolds, Mathematica Pannonica, 4/2 (1993), 191-203.

[E-K 2] S. Ewert-Krzemieniewski, On manifolds with curvature condition of recurrent type of the second order, Period. Math. Hungar. 34 (1997), no. 3, 185–194.

[K] G. J. Kruchkovich, O prostranstvah V. F. Kagana, in: V. F. Kagan, Subproektivnye prostranstva, Gosudarstvennoe Izdatelstvo Fiziko–Matematicheskoi Literatury, Moskva,1961 (in Russian).

[P] M. Prvanovć, Extended recurrent manifolds, (Russian), Izv. Vyssh. Uchebn. Zaved. Mat. 1999, no. 1, 41–50; translation in Russian Math. (Iz. VUZ) 43 (1999), no. 1, 38–47.

[W] A. G. Walker, On Ruse’s spaces of recurrent curvature, Proc. Lond. Math. Soc., 52 (1950), 36-64.

Institute of Mathematics Technical University of Szczecin al. Piastów 17, 70-310 Szczecin POLAND e-mail: ewert@arcadia.tuniv.szczecin.pl

FIRST–ORDER DIFFERENTIAL INVARIANTS OF THE SPLITTING SUBGROUPS OF THE POINCARÉ GROUP P (1, 4)

V. M. FEDORCHUK and V. I. FEDORCHUK

Abstract

The differential invariants of the Lie groups of the point transformations play an important role in geometry (see, for example, [1]), group analysis of differential equations (see, for example, [1–3]), etc.

Our communication is devoted to the construction of the differential invariants for nonconjugate subgroups of the generalized Poincaré group P (1, 4). The group P (1, 4) is a group of rotations and translations of the five–dimensional Minkowski space M(1, 4). For all splitting subgroups of the group P (1, 4), the first–order differential invariants have been found.

The results obtained have been used for the construction of the first-order differential equations in the space M(1, 4) × R(u), which are invariant under splitting subgroups of the group P (1, 4).

Since the Lie algebra of the group P (1, 4) contains as subalgebras the Lie algebra of the Poincaré group P (1, 3) and the Lie algebra of the extended Galilei group G̃(1, 3) [ 4 ], the obtained differential equations can be used in relativistic and non–relativistic physics.

1. Olver P.J., Applications of the Lie Groups to Differential Equations, Springer-Verlag, New York, 1986.

2. Lie S., Transformationsgruppen, In: 3 Bd, Leipzig, 1893, Bd 3.

3. Ovsiannikov L.V., Group Analysis of Differential Equations, Academic Press, New York, 1982.

4. Fushchych W.I. and Nikitin A.G., Symmetries of Equations of Quantum Mechanics, Allerton Press Inc., New York, 1994.

V. M. FEDORCHUK Institute of Mathematics, Pedagogical Academy PL – 30–084 Kraków, ul. Podchora̧żych 2 POLAND and Pidstryhach Institute of Applied Problems of Mechanics and Mathematics National Ukrainian Academy of Sciences 3b Naukowa St, 79–053, Lviv UKRAINE e-mail: vas_fedorchuk@yahoo.com

V. I. FEDORCHUK Ivan Franko Lviv National University 1 Universytetska St, Lviv, 7900 UKRAINE

LIE BRACKET ON AFFINE BUNDLES

JANUSZ GRABOWSKI, KATARZYNA GRABOWSKA, PAWEŁ URBAŃSKI

Abstract

Natural analogs of Lie brackets on affine bundles are studied. In particular, a close relation to Lie algebroids and duality with certain affine analog of Poisson structures is established as well as affine versions of the complete lift and the Cartan exterior calculus.

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

Katarzyna GRABOWSKA Faculty of Physics University of Warsaw Hoża 74, 00–682 Warszawa POLAND e-mail: koniecz@fuw.edu.pl

Paweł URBAŃSKI Faculty of Physics University of Warsaw Hoża 74, 00–682 Warszawa POLAND e-mail: urbanski@fuw.edu.pl

THE WORKS OF PROFESSOR STANISŁAW GOŁĄB IN THE FIELD OF FUNCTIONAL EQUATIONS

ROMAN GER

Abstract

Although the impressive scientific output of Professor Stanisław Gołąb uniquely qualifies him as the very prominent Polish geometrist even an extremely brief (just one sentence altogether) biographic note in Nowa encyklopedia powszechna PWN, vol. 2, p. 567, states that S. Gołąb’s papers deal also with functional equations. Actually, approximately 20% of his scientific publications might be classified among this theory. The list of 50 papers presented below (I will be frequently referring the listeners to this list during my talk) contains the items that are more or less directly related to functional equations. Among them the monograph [26] written jointly with János Aczél, the world-wide leader in this field, belongs to one of the first books devoted entirely to functional equations. The fundamental one among those considered in [26] is the translation equation

F (F (x, s), t) = F (x, ts)

intensively studied throughout several decades and nowadays. This equation, occurring while studying the problem of determination of geometric objects, yields also the subject of numerous papers listed below.

Another equation of basic importance is commonly known as the Gołąb-Schinzel functional equation

f(x + yf(x)) = f(x)f(y)

which was obtained in [24] while looking for some special subgroups or subsemigroups of a given continuous group of transformations. Enormous number of mathematicians were and are inspired by that idea which led them to substantial generalizations of this equation studied in various classes of functions and pretty general spaces.

Equally inspiring was the paper [21] written jointly with Mieczysław Kucharzewski who, like his teacher S. Gołąb, had great achievements in applying functional equation methods to geometry. Studying homomorphisms of the differential groups L_{n}^{s} and, in particular, the multiplicative Cauchy equation involving matrix arguments is extremely vivid up to now.

The other topics that will be discussed and reported on may be framed as follows:

• characterizations of mappings,

• other functional equations occurring in geometry,

• functional equations stemming from algebra,

• functional equations stemming from analysis,

• the significance of the domain of functional equation and the idea of local solutions,

• the functional equation of brigade;

plainly, all of them dealt with by Professor Stanisław Gołąb in his papers presented below.

References

[1] Sur les fonctions homogènes, I. Équation d’Euler, C.R. Soc. Sci. Varsovie 25 (1932), 105-110.

[2] Sur la métrique angulaire des espaces généraux.I. Le pricipe de Chasles, Wiadom. Mat. 38 (1933), 1-12.

[3] Sur une définition axiomatique des nombres conjugués pour les nombres complexes ordinaires, Opusc. Math. (Cracov) 1 (1937), 1-11.

[4] Sur la possibilité de la détermination de “coniugum” dans le corps des nombres hypercomplexes à deux unités, Rev. Ci. (Lima) 40 (1937), 21-30.

[5] Sur quelques points concernant la notion du comitant, Ann. Soc. Polon. Math. 17 (1938), 177-192.

[6] Über die Klassifikation der geometrischen Objekte, Math. Z. 44 (1938), 104-114.

[7] Über eine Funktionalgleichung der Theorie der geometrischen Objekte, Wiadom. Mat. 45 (1938), 97-137.

[8] Sur la théorie des objets géométriques (Les objets diffèrentiels purs de deuxième et de troisième classe), Ann. Soc. Polon. Math. 19 (1946), 7-35.

[9] Sur la théorie des objets géométriques (Réduction de objets géométriques spéciaux de premiére classe aux objets du type Δ), Ann. Soc. Polon. Math. 20 (1947), 10-27.

[10] Sur la notion de dèrivée covariante, Colloq. Math. 1 (1948), 160.

[11] Sur les objets géométriques non-différentiels, Bull. Acad. Polon. Sci. Lettr. Cl. Sci. Math. Natur. Sér. A (1949), 67-72.

[12] Contribution à la théorie des objets géométriques, Prace Mat.-Fiz. (Warszawa) 47 (1949), 1-15.

[13] Sur les objets géométriques à une composante, Ann. Soc. Polon. Math. 23 (1950), 79-89.

[14] Sur la dèrivée covariante des objets géométriques de deuxième classe, Ann. Polon. Math. 1 (1954), 107-113.

[15] Über den Begriff der kovarianten Ableitung, Nieuw. Arch. Wisk. 2 (1954), 90-96.

[16] Géométrie différentielle vis-à-vis des hypothèses d’une faible régularité, Rev. Roumaine Math. Pures Appl. 1 (1956), 99-112.

[17] Zum distributiven Gesetz der reellen Zahlen, Studia Math. 15 (1956), 353-358.

[18] Un théorème sur la valeur moyenne θ dans la formule des accroissements finis, Ann. Polon. Math. 1 (1956), 118-125 (joint work with S. Łojasiewicz).

[19] Sur l’équation fonctionnelle f(X · Y) = f(X) · f(Y), Colloq. Math. 4 (1957), 265.

[20] Sur la l’algèbre des objets géométriques de première classe à une composante), Ann. Polon. Math. 4 (1957), 226-248 (joint work with H. Pidek).

[21] Über den Begriff der Pseudogröβen, Tensor N.S. 8 (1958), 78-89 (joint work with M. Kucharzewski).

[22] Sur l’équation f(X · Y) = f(X) · f(Y), Ann. Polon. Math. 6 (1959), 1-13.

[23] Zagadnienia teorii obiektów geometrycznych, Prace Mat. 3 (1959), 55-67.

[24] Sur l’équation fonctionnelle f[x+y ·f(x)] = f(x)·f(y), Publ. Math. Debrecen 6 (1959), 113-125 (joint work with A. Schinzel).

[25] Sur les comitants algèbriques des tenseurs, Ann. Polon. Math. 9 (1960), 113-118.

[26] Funktionalgleichungen der Theorie der geometrischen Objekte, Monografie Mat. 39, PWN Warszawa, 1960 (a book written jointly with J. Aczél).

[27] On some new geometrical interpretations of the torsion of a skew curve, Ann. Polon. Math. 7 (1960), 269-278 (joint work with J. Kordylewski and M. Kuczma).

[28] Das Doppelverhältnis als Lösung einer Funktionalgleichung, Ann. Polon. Math. 9 (1960), 183-187 (joint work with J. Aczél, M. Kuczma and E. Siwek).

[29] Sur l’équivalence des objets géométriques de deuxième classe dont le nombre de composantes est ègal à la dimension de l’espace, Czechoslovak Math. J. 11 (1961), 475-479.

[30] La relation d’équivalence et les objets géométriques, Fund. Math. 50 (1962), 381-386.

[31] Über ein Funktionalgleichungssystem von O.E. Gheorgiu, Tagungsbericht “Funktionalgleichungen”, 7-11 Okt. 1963, Mathem. Forschungsinstitut Oberwolfach, pp. 9-10.

[32] Einige grundlegende Fragen der Theorie der Funktionalgleichungen, Glasnik Mat.-Fiz.. Astronom 20 (1965), 58-63.

[33] Über die Funktionalgleichung der Funktion Arccosinus, I. Die lokalen Lösungen, Publ. Math. Debrecen 12 (1965), 159-174 (joint work with L. Losonczi).

[34] La comparaison de la rapidité de convergence des approximations successives de la méthode de Newton avec la méthode de “regula falsi”, Mathematica (Cluj) 8 (1966), 45-49.

[35] Über ein System von Funktionalgleichungen, Ann. Polon. Math. 17 (1966), 223-243 (joint work with O.E. Gheorgiu).

[36] Über die Funktionalgleichung der Funktion Arccosinus, II, Publ. Math. Debrecen 13 (1966), 183-185 (joint work with L. Losonczi).

[37] Über Differentialkomitanten ersten Ordnung, I, Colloq. Math. 16 (1967), 173-184.

[38] Über Differentialkomitanten ersten Ordnung, II, Ann. Polon. Math. 20 (1968), 81-89.

[39] Über die Grundlagen der affinen Geometrie, Jber. Deutsch. Math.-Verein. 71 (1969), 138-155.

[40] Über ein Funktionalgleichungssystem aus dem Gebiete der Geometrie, I Intern. Kolloquium über Funktionalgleichungen und ihre Anwendungen, Publ. of the Technical University, Miskolc (1970), 237-242.

[41] Functional equations in geometry, Zeszyty Nauk. Uniw. Jagiellońskiego. Prace Mat. 14 (1970), 13-19.

[42] Remarks on one-parameter subsemigroupes of the affine group and their homo- and isomorphisms, Aequationes Math. 4 (1970), 1-10 (joint work with J. Aczél).

[43] Über das Flächenmass in Vektorräumen, Prace Mat. 14 (1970), 51-57 (joint work with M. Kucharzewski).

[44] Les équations fonctionnelles et la théorie des groupes, Rev. Roumaine Math. Pures Appl. 15 (1970), 1395-1406 (joint work with Z. Krzeszowiak).

[45] Sur l’équation fonctionnelle des brigades, [Corso tenuto a La Mendola (Trento) dal 20 al 28 agosto 1970], Edizioni Cremonese, Roma, 1971, pp. 143-151.

[46] Note on inner product in vector spaces, Aequationes Math. 8 (1972), 74-75 (joint work with H. Światak).

[47] Ein Beitrag zur Axiomatisierung des Begriffs des Skalarproduktes, Demonstratio Math. 5 (1973), 273-288.

[48] Sur un systéme d’équations fonctionnelles lié au rapport anharmonique, Ann. Polon. Math. 29 (1974), 273-280.

[49] Theorie der Quasigruppen, [In: Grundlagen der Geometrie und algebraische Methoden, Internat. Kolloq., Pädagog. Hochsch. “Karl Liebknecht”, Potsdam, 1973], Potsdamer Forschungen, Reihe B, Heft 3, Potsdam, 1974, pp. 137-151.

[50] Über das Carnotsche Skalarprodukt in schwach normierten Vektorräumen, Aequationes Math. 13 (1975), 9-13.

Institute of Mathematics Silesian University Bankowa 14, 40-007 Katowice POLAND

 http://www.imi.wsp.czest.pl/pracownicy/ger.html

e-mail: romanger@us.edu.pl

SYMPLECTIC STRUCTURES AND DIFFEOMORPHISMS

BOGUSŁAW HAJDUK

Abstract

I discuss the problem of existence of symplectic structures on tori with different differential structures and related question of calculating the action of the group of diffeomorphisms on the homotopy classes of almost complex structures. I show that diffeomorphisms which are topologically isotopic to the identity may act nontrivially.

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

DUALITY OF SYMMETRIES IN 4–DIMENSIONAL LORENTZ MANIFOLDS^{2}

GRAHAM S. HALL

Abstract

This paper presents a brief discussion of the description of symmetries in 4-dimensional Lorentz manifolds (with a view to the space-time of general relativity). The orbit structure in terms of foliations is particularly stressed. The main symmetry discussed is local isometry, but other symmetries are briefly mentioned.

1 Introduction

The aim of this paper is to present a brief, reasonably modern approach to the study of symmetry in general relativity theory, that is, on a 4-dimensional manifold admitting a Lorentz metric. Throughout, M will be a smooth, connected, Hausdorff manifold admitting a smooth, Lorentz metric g of signature (−, +, +, +) (and hence M is paracompact [1]). If m ∈ M, T_{m}M will denote the tangent space to M at m. A Lie derivative is denoted by L. When component notation is used, a partial derivative and a covariant derivative with respect to the Levi-Civita connection Γ associated with g are denoted, respectively, by a comma and a semi-colon.

In Einstein’s general relativity theory, M plays the role of the space-time and the geometrical objects g, Γ and the curvature tensor on M derived from Γ collectively describe the gravitational field. Einstein’s equations provide the physical restrictions on these objects. However, they will not be required in this paper.

Of course, there are many different types of symmetry studied in general relativity, for example, (local) isometries, homotheties, conformal isometries, affine and projective collineations and symmetries of the curvature and related tensors (for reviews see [2, 3]). The purpose of this paper, however, is more general, and will concentrate on techniques rather than the specific symmetry involved. Nevertheless, local isometries will finally be studied as an application. So far as the present author is aware, the mathematical study of symmetry in general relativity theory has not taken into account the progress made in the recent studies of the integrability of vector fields and foliations. The main purpose of this paper is to attempt a small step in this direction and to set on a more rigorous basis the general theory of symmetries and their associated orbits.

2 Space-Time Geometry and Decomposition

Let m ∈ M and 0 ≠ v ∈ T_{m}M. Then v is called spacelike (respectively, timelike or null) if g(v, v) > 0 (respectively, g(v, v) < 0 or g(v, v) = 0). A 1-dimensional subspace of T_{m}M is called a direction (at m), and is referred to as a spacelike (respectively, timelike or null) direction if it is spanned by a spacelike (respectively, timelike or null) vector at m. If U is a 2-dimensional subspace of T_{m}M, then U is called spacelike (respectively, timelike or null) if all non-zero members of U are spacelike (respectively, if U contains exactly two distinct null directions or if U contains exactly one null direction). If U is a 3-dimensional subspace of T_{m}M, the same definitions as in the 2-dimensional case apply except that in the timelike case, one insists that at least two (or, equivalently, infinitely many) distinct null directions are contained in U. These definitions are exclusive and exhaustive of all non-zero members of T_{m}M and all 1-, 2- and 3-dimensional subspaces of T_{m}M. A (smooth) submanifold N of M of dimension 1, 2 or 3 is called spacelike at m ∈ M if its tangent space is a spacelike direction or subspace of T_{m}M and spacelike if it is spacelike at each m ∈ M (and similarly for timelike and null). If N is a spacelike (respectively, timelike) submanifold of M, then g induces a positive definite (respectively, Lorentz) metric on N.

2This is the text of an invited lecture given at the 3rd Conference GEOMETRY and TOPOLOGY of MANIFOLDS held in Krynica, Poland, May 2001.

It should be pointed out here that the term (smooth) submanifold of M means what is sometimes referred to as a (smooth) immersed submanifold of M. Thus, if M′ is a submanifold of M, then M′ is a subset of M which has a manifold structure, and is such that the inclusion map i : M′ → M is a (smooth) immersion. If, in addition, the manifold topology (from the manifold structure) on M′ equals its subspace topology as a subspace of M when the latter has its manifold topology, then M is called a regular or embedded submanifold. One of the advantages of regular submanifolds is that if M_{1} and M_{2} are smooth manifolds and f : M_{1} → M_{2} is a smooth map whose range f(M_{1}) lies inside a smooth regular submanifold N_{2} of M_{2}, then the map f : M_{1} → N_{2} is also smooth. If N_{2} is not regular, this latter map may not even be continuous (but if it is continuous then f : M_{1} → N_{2} is smooth). There is a type of submanifold introduced, as far as the author is aware, by Stefan [4, 5], and which is intermediate between submanifolds and regular submanifolds. A leaf of M is a connected (immersed) submanifold N of M with the additional property that, if T is any locally connected topological space, and f : T → M is a continuous map whose range lies inside N, then the map f : T → N is continuous. It follows [4] that if M_{1} and M_{2} are smooth manifolds and N_{2} is a leaf of M_{2}, and f : M_{1} → M_{2} is a smooth map whose range lies in N_{2}, then the map f : M_{1} → N_{2} is continuous, and hence smooth. If N is a subset of M admitting two structures N_{1} and N_{2} as smooth regular submanifolds of M, then, from earlier remarks in this paragraph, the identity maps N_{1} → N_{2} and N_{2} → N_{1} are each smooth and so N_{1} = N_{2} and the regular submanifold structure is unique (see, e.g. [6]). The same uniqueness conclusion also holds if regular submanifold is replaced by leaf [4]. Clearly, every connected regular submanifold is a leaf, but the three types of (connected) submanifold structures (immersed, embedded and leaf) are distinct since the irrational wrap on the torus is a leaf which is not regular [4], whilst the well known figure of eight in ℝ^{2} (see, e.g. [6]) is a connected submanifold which is easily shown not to be a leaf.

Now let A be a vector space of global, smooth vector fields on M and define the distribution Δ on M associated with A by [7]

m → Δ(m) = {X(m) : X ∈ A} ⊆ T_{m}M.  (2.1)

Then, for i = 0, 1, 2, 3, 4 and p = 1, 2, 3, define subsets V_{i}, S_{p}, T_{p} and N_{p} by

V_{i} = {m ∈ M : dim Δ(m) = i}  (2.2)

S_{p} = {m ∈ M : dim Δ(m) = p and Δ(m) is spacelike}  (2.3)
T_{p} = {m ∈ M : dim Δ(m) = p and Δ(m) is timelike}
N_{p} = {m ∈ M : dim Δ(m) = p and Δ(m) is null}

Thus, M = ∪_{i=0}^{4}V_{i} and V_{p} = S_{p} ∪ T_{p} ∪ N_{p} (p = 1, 2, 3). This decomposition of M can be refined topologically by appealing to the rank theorem to see that M = ∪_{i=k}^{4}V_{i} is open in M for k = 0, …, 4. This can then be used to reveal the following disjoint decompositions of M [7]

M = V_{4} ∪ ⋃_{i=0}^{3}V_{i} ∪ Z_{1}  (2.4)

M = V_{4} ∪ ⋃_{p=1}^{3}S_{p} ∪ ⋃_{p=1}^{3}T_{p} ∪ ⋃_{p=1}^{3}N_{p} ∪ V_{0} ∪ Z  (2.5)

where denotes the topological interior (and V_{4} = V_{4}) and where Z and Z_{1} are closed subsets of M each with empty interior.

3 Local Space-Time Symmetries

With A as in the last section, let A_{1}, …, A_{k} ∈ A and let ϕ_{t_{1}}^{1}, …, ϕ_{t_{k}}^{k} be the smooth, local diffeomorphisms associated with them, for appropriate values of t. Then consider the set of all such local diffeomorphisms (where defined) of the form

m → ϕ_{t_{1}}^{1}(ϕ_{t_{2}}^{2}(⋯ ϕ_{t_{k}}^{k}(m) ⋯)), (m ∈ M)  (3.1)

for each choice of k, X_{1}, ⋯, X_{k} and admissible (t_{1}, ⋯, t_{k}) ∈ ℝ^{k}. There is an equivalence relation on M given by m_{1} ∼ m_{2} if some local diffeomorphism of the form (3.1) maps m_{1} into m_{2}. The associated equivalence classes in M are called the orbits of A and it is known that these orbits can each be given the structure of a connected, smooth submanifold of M [8, 4, 5]. In fact, Stefan has shown that these submanifolds constitute a foliation with singularities, so that each has the extra property of being a leaf. He also showed that if O is any such leaf and m ∈ O, then the tangent space to O at m is the subspace {f_{∗}v : v ∈ Δ(m′)} of T_{m}M for each f of the form (3.1) and each m′ ∈ M such that f(m′) = m. This subspace need not equal Δ(m). The condition that it does so for each m ∈ M is equivalent to the condition that the orbits are integral manifolds of the set A and then A is integrable [8, 4, 5].

In general relativity, the situations of interest occur when A is a Lie algebra (under the Lie bracket operator) of global, smooth vector fields on M and then attention is directed to the nature of the orbits of the symmetries represented by A and whether they are integral manifolds of A. If dim Δ(m) is constant on M, the Fröbenius theorem (see e.g. [6]) guarantees that the orbits are submanifolds and, in fact, integral manifolds of A. The work of Stefan then ensures that the orbits are leaves of a foliation on M. If dim Δ(m) is not constant, then integrability need not follow. If, however, A satisfies the locally finitely generated condition (i.e. that each m ∈ M has an open neighbourhood U and a finite subset A′ of A such that each X ∈ A, when restricted to U, is a combination of members of A′ (restricted to U) with coefficients which are smooth maps U → ℝ), then Hermann [9] has shown that A is integrable (in fact, he showed more than this). Thus, if A is a finite-dimensional Lie algebra, it is integrable and, again [4, 5], the orbits are leaves of a foliation with singularities.

The symmetries usually studied in general relativity are described by a Lie algebra of global, smooth vector fields on the space-time M, with each particular symmetry being characterised by insisting upon the appropriate property being possessed by the resulting local diffeomorphisms of the type (3.1) (see, e.g. [2, 3]). Thus, projective symmetry is defined by insisting that each map (3.1) takes geodesics to geodesics and the resulting Lie algebra A, now labelled P (M), is the set of all global, smooth vector fields on M with this property. The vector fields in P (M) are called projective and are characterised by the condition that, in any chart of M

X_{a;b} = \frac{1}{2}h_{ab} + F_{ab}  (h_{ab} = h_{ba}, F_{ab} = −F_{ba})  (3.2)
h_{ab;c} = 2g_{ab}ψ_{c} + g_{ac}ψ_{b} + g_{bc}ψ_{a}

for some closed 1-form field ψ and 2-form field F on M. Special cases are the affine vector fields (for which ψ ≡ 0 on M and whose associated maps (3.1) preserve also the geodesic affine parameter), the homothetic vector fields (which are affine and satisfy h_{ab} = cg_{ab}, c ∈ ℝ) and the Killing vector fields which are homothetic with c = 0 and so L_{X}g = 0 (and for which each map (3.1) is a local isometry). The sets of all affine, homothetic and Killing vector fields on M are labelled A(M), H(M) and K(M) respectively, and K(M) ⊆ H(M) ⊆ A(M) ⊆ P (M), with each being a subalgebra of P (M). Conformal symmetry is defined by insisting that each map f in (3.1) is a local conformal diffeomorphism, that is, f^{∗}g = αg for some appropriate local, smooth real valued function α. The resulting set of all global, smooth vector fields on M with this property is labelled C(M) and its members are called conformal. Then X ∈ C(M) is characterised in any chart of M by

X_{a;b} = ϕg_{ab} + F_{ab}  (F_{ab} = −F_{ba})  (3.3)

where ϕ : M → ℝ and F is a 2-form field on M. The set C(M) is a Lie algebra and H(M) and K(M) above are subalgebras of it. Now it is well-known that P (M) and C(M) are finite-dimensional with dim P (M) ≤ 24 and dim C(M) ≤ 15 and so it follows from the discussion above that the orbits of P (M) and C(M) are each foliations with singularities and are integral manifolds of P (M) and C(M), respectively, and similarly for their subalgebras mentioned above. [It is remarked that the local action on M provided by the local diffeomorphisms described in the above Lie algebras need not lead to a global Lie group action on M. This occurs if and only if each vector field in the Lie algebra is complete [10].]

4 The Killing Algebra K(M)

Consider the finite-dimensional Lie algebra of Killing vector fields K(M) on M. The material of section 3 shows that the orbits associated with K(M) are leaves of a foliation with singularities and are integral manifolds of K(M). It also shows that, if O is any orbit of K(M), and f any associated local isometry of K(M) whose domain and range are the open subsets U and U′ of M, then f gives rise to a smooth map U ∩ O → U′ whose range lies in the leaf O. Hence, it gives rise to a smooth map U ∩ O → U′ ∩ O, since U′ ∩ O is an open and hence, regular submanifold of O. Then if m ∈ U ∩ O, f_{∗}(T_{m}O) = T_{f(m)}O. The definitions at the beginning of section 2 then show that, since f is a local isometry, O is either spacelike, timelike or null. If O is spacelike (respectively, timelike), then g induces a metric h = i^{∗}g on O which is positive-definite (respectively, Lorentz). If X ∈ K(M) then X is tangent to O and so there is a unique smooth, global vector field X̃ on O such that i_{∗}X̃ = X. If O is non-null with induced metric h, then the condition that X ∈ K(M), that is L_{X}g = 0, is easily shown to imply that LX̃h = 0 and so X̃ is a Killing vector field on O with metric h, that is, X̃ ∈ K(O). In fact, the map k : X → X̃ is a Lie algebra homomorphism K(M) → K(O).

In general, the map k is neither injective nor surjective. That the map k is not surjective can be seen from the space-time metric given in a global chart on {(x, y, z, t) ∈ ℝ^{4} : t > 0} ≡ M by

ds^{2} = −dt^{2} + tdx^{2} + e^{2t}dy^{2} + e^{3t}dz^{2}.  (4.1)

Here K(M) is 3-dimensional, being spanned by the vector fields \frac{∂}{∂x}, \frac{∂}{∂y} and \frac{∂}{∂z}. However, each subset O of constant t is an orbit of K(M) and is, with its induced metric, flat Euclidean 3-space and so dim K(O) = 6.

To investigate whether k is injective or not, let 0 ≠ X ∈ K(M) and let m ∈ M with X(m) = 0. Then the local isometries ϕ_{t} associated with X satisfy ϕ_{t}(m) = m and m is called a zero of X (or a fixed point of each ϕ_{t}). If U is a coordinate neighbourhood of m with coordinates y^{a}, then the linear isomorphism ϕ_{t∗} : T_{m}M → T_{m}M is represented in the basis (\frac{∂}{∂y^{a}})_{m} by the matrix

e^{tB} = exp t(\frac{∂X^{a}}{∂y^{b}} )_{m}  (4.2)

where B_{b}^{a} ≡ (\frac{∂X^{a}}{∂y^{b}} )_{m} is the linearisation of X at m. Thus, since X ∈ K(M), it follows from (3.2) that B_{b}^{a} = (F^{a}_{b})_{m}. Also, since X is affine, if χ is the usual exponential diffeomorphism from some open neighbourhood of 0 ∈ T_{m}M onto some open neighbourhood V of m, then [11]

ϕ_{t} ∘ χ = χ ∘ ϕ_{t∗}.  (4.3)

It is easily checked from this that, in the resulting normal coordinate system x^{a} with domain V about m, the components X^{a} of X are linear functions of the coordinates x^{a}. Since B_{b}^{a} = (F^{a}_{b})_{m} is skew self- adjoint with respect to g(m), it follows that the rank of B is even. If B = 0 then X ≡ 0 on M and so B has rank 2 or 4. The zeros of X in V have coordinates satisfying B^{a}_{b}x^{b} = 0 and so, if B = 4, the zero m is isolated, whereas if B = 2, the zeros of X in V can be given the structure of a 2-dimensional, regular submanifold N of the open submanifold V [12, 13]. Now return to the map k and suppose it is not injective. Let O be the orbit of K(M) through m. Then there exists X ∈ K(M), X ≢ 0, such that X vanishes on O, that is, X̃ = 0. Since m is thus not isolated, A = 2, and so the zeros of X in V are exactly the points on the 2-dimensional regular submanifold N of V. Let O′ = O ∩V. Then O′ is an open subset (and hence an open submanifold) of O. It follows that O′ is a submanifold of M contained in the open (hence regular) submanifold V of M and hence O′ is a submanifold of V [6]. But then O′ ⊆ N ⊆ V, with O′ and N submanifolds of V with N regular. It follows that O′ is a submanifold of N and so dim O′ ≤ dim N and hence dim O(= dim O′) ≤ 2. Hence, if dim O is 3 or 4, k is injective. If, however, dim O ≤ 2, k can fail to be injective, as the following example shows. Let M_{1} and M_{2} be 2-dimensional, connected, smooth manifolds with M_{2} = ℝ^{2}. Let g_{1} be a positive definite metric on M_{1} with K(M_{1}) 1-dimensional and spanned by a Killing vector field with a single zero at m ∈ M_{1}. Let g_{2} be the usual Minkowski metric on M_{2}, so that dim K(M_{2}) = 3. Then the spacetime M_{1} × M_{2} with metric g_{1} ⊗ g_{2} is such that dim K(M) = 4 and O = {m_{1}} × M_{2} is a 2-dimensional, timelike orbit of K(M) with dim K(O) = 3. Thus, the map K(M) → K(O) is not injective.

If O is an orbit of K(M), it was pointed out above that O is either spacelike, timelike or null. Thus, if dim O = p (1 ≤ p ≤ 3) and O ∩ S_{p} ≠ ∅, then O ⊆ S_{p} (and similarly for T_{p} and N_{p}). It is convenient at this point to distinguish between orbits which are, in some sense, stable with respect to their type and dimension and those which are not. Thus, an orbit is called stable if it is contained in one of the subsets S_{p}, T_{p} or N_{p}(1 ≤ p ≤ 3). Actually, since the inner product of a Killing vector field and the tangent vector to an affinely parameterised geodesic is constant along the geodesic, an argument based on the normal geodesics to orbits contained in S_{3} and T_{3} and an appeal to the rank theorem similar to that made at the end of section 2 shows that S_{3} and T_{3} are open. Thus, all orbits in S_{3} and T_{3} are stable. Regarding the stability of orbits, it is easy to show that, if O is any orbit of K(M) such that O ∩ S_{p} ≠ ∅ (1 ≤ p ≤ 3), then O ⊆ S_{p} (and similarly for T_{p} and N_{p}). It is now possible to prove a number of results about how the existence of a certain type of stable orbit restricts the dimension of K(M). These results are often used in the relativistic literature without justification. Some similar (but, as yet, incomplete) results are available in a similar context for unstable orbits [14].

In summary then (see [14, 15] for further discussion), the Lie algebra K(M) of global, smooth Killing vector fields on a space-time M with smooth, Lorentz metric g is finitedimensional and the orbits resulting from the maps (3.1) constitute a foliation with singularities. The maps (3.1) are smooth (local) maps M → M (and also O → O, for any orbit O) and give rise to a Lie group (global) action on M if and only if each member of K(M) is complete. A convenient decomposition of M with respect to the Lorentz metric g on M is provided by (2.2)–(2.5). The tangency of the members of K(M) to an orbit leads to a natural Lie algebra homomorphism K(M) → K(O) which is easily seen to be not necessarily surjective and which is, perhaps less obviously, not necessarily injective, but is injective if dim O ≥ 3. This latter remark stems from a study of the zeros of the members of K(M). The orbits of K(M) were then divided into stable and unstable ones and the known (and used) results in orbit theory in general relativity can then be shown to apply to the stable orbits.

5 Acknowledgements

The author wishes to thank Profs Robert Wolak and Jan Kubarski for their hospitality in Poland and for several useful discussions. He also acknowledges many valuable discussions with Profs Dimitri Alexeevski and Janos Szenthe.

References

[1] R. Geroch. J. Math. Phys., \underline{9}, (1968), 1739.

[2] A. V. Aminova. J. Soviet. Maths., \underline{55}, (1991), 1995.

[3] G. S. Hall. Grav. and Cosmology, \underline{2}, (1996), 270.

[4] P. Stefan. Proc. London. Math. Soc., \underline{29}, (1974), 699.

[5] P. Stefan. J. London. Math. Soc., \underline{21}, (1980), 544.

[6] F. Brickell and R. S. Clark. Differentiable Manifolds. Van Nostrand, London, 1970.

[7] G. S. Hall. The Petrov Lectures, to appear in the proceedings of the meeting Volga XII, Kazan, (2000).

[8] H. J. Sussmann. Trans. Am. Math. Soc., \underline{180}, (1973), 171.

[9] R. Hermann. International Symposium on non-linear differential equations and nonlinear mechanics. New York, Academic press, (1963), 325.

[10] R. S. Palais. Mem. Am. Math. Soc., No. 22, (1957).

[11] S. Kobayashi and K. Nomizu. Foundations of Differential Geometry (Vol. 1). Interscience, New York, 1963.

[12] G. S. Hall. Gen. Rel. Grav., \underline{20}, (1988), 671.

[13] G. S. Hall. J. Math. Phys., \underline{31}, (1990), 1198.

[14] G. S. Hall. Preprint, University of Aberdeen, (2000).

[15] G. S. Hall. Gen. Rel. Grav., \underline{30}, (1988), 1099.

Department of Mathematical Sciences University of Aberdeen Aberdeen AB24 3UE Scotland UNITED KINGDOM e-mail: g.hall@maths.abdn.ac.uk

HOLONOMY THEORY AND 4–DIMENSIONAL LORENTZ MANIFOLDS

GRAHAM S. HALL

Abstract

Let M be a smooth, 4–dimensional, connected, Hausdorff manifold with smooth Lorentz metric g. Let Φ be the holonomy group of M associated with the Levi–Civita connection from g with holonomy algebra ϕ. The purpose of this talk is to give some results regarding the possibilities for Φ. Let L be the Lorentz group with connected component L_{0} and with Lie algebra L. Clearly, Φ is a subgroup of L and the well-known classification of L is thus important in this work. Φ is a Lie group and, if M is simply connected, Φ is a connected subgroup of L_{0}. In this case, since ϕ is a subalgebra of L, there exists a one–to–one correspondence between the possibilities for Φ and the subalgebras of L. This gives a classification up to isomorphism of the possible holonomy groups of M. A coarser classification can (and will) be given in terms of covariantly constant and recurrent vector fields on M. Another useful approach (thinking of the pair (M, g) as the space–time of Einstein’s general relativity) is to seek the possibilities for Φ when the energy–momentum tensor representing the physics of space–time is given. This will be discussed for the more commonly used energy–momentum tensors. Some remarks will also be made concerning the use of holonomy theory in the study of space–time curvature structure and space–time symmetries and in the theory of the Petrov classification of the Weyl tensor for gravitational fields.

Department of Mathematical Sciences

University of Aberdeen

Aberdeen AB24 3UE

Scotland

UNITED KINGDOM

e-mail: g.hall@maths.abdn.ac.uk

ON HYPERSURFACES WITH TYPE NUMBER TWO IN SPACE FORMS

MARIAN HOTLOŚ

Abstract

1. Basic notations

Let (M, g), n ≥ 3, be a connected semi-Riemannian manifold of class C^{∞}. We denote by ∇, R, C, S and κ the Levi-Civita connection, the Riemann-Christoffel curvature tensor, the Weyl conformal curvature tensor, the Ricci tensor and the scalar curvature of (M, g), respectively. The Ricci operator S is defined by g(SX, Y) = S(X, Y), where X, Y ∈ Ξ(M), Ξ(M) being the Lie algebra of vector fields on M. Next, we define the endomorphisms R(X, Y), C(X, Y) and X ∧_{A} Y of Ξ(M) by

(X ∧_{A} Y)Z = A(Y, Z)X − A(X, Z)Y ,
R(X, Y)Z = [∇_{X}, ∇_{Y}]Z − ∇_{[X,Y]}Z ,
C(X, Y)Z = R(X, Y)Z − \frac{1}{n − 2} (X ∧_{g} SY + SX ∧_{g} Y

−\frac{κ}{n − 1} X ∧_{g} Y)Z ,

respectively, where A is a symmetric (0, 2)-tensor and X, Y, Z ∈ Ξ(M). The Riemann-Christoffel curvature tensor R, the Weyl conformal curvature tensor C and the (0, 4)-tensor G of (M, g) are defined by

R(X_{1}, X_{2}, X_{3}, X_{4}) = g(R(X_{1}, X_{2})X_{3}, X_{4}) ,
C(X_{1}, X_{2}, X_{3}, X_{4}) = g(C(X_{1}, X_{2})X_{3}, X_{4}) ,
G(X_{1}, X_{2}, X_{3}, X_{4}) = g((X_{1} ∧_{g} X_{2})X_{3}, X_{4}) ,

respectively. For a (0, k)-tensor field T, k ≥ 1, and a (0, 2)-tensor field A on (M, g) we define the tensors R · T and Q(A, T) by

(R · T)(X_{1}, …, X_{k}; X, Y) = −T (R(X, Y)X_{1}, X_{2}, …, X_{k})
− ⋯ − T (X_{1}, …, X_{k−1}, R(X, Y)X_{k}) ,
Q(A, T)(X_{1}, …, X_{k}; X, Y) = −T ((X ∧_{A} Y)X_{1}, X_{2}, …, X_{k})
− ⋯ − T (X_{1}, …, X_{k−1}, (X ∧_{A} Y)X_{k}) ,

respectively. Putting in the above formulas T = R, T = S or T = C, A = g or A = S, we obtain the tensors R · R, Q(g, R), Q(S, R), R · S, Q(g, S), R · C, Q(g, C) and Q(S, C). We define the following subsets of M: U_{R} = \{x ∈ M | R − \frac{κ}{(n−1)n} G ≠ 0 at x\}, U_{S} = \{x ∈ M | S − \frac{κ}{n} g ≠ 0 at x\}, U_{C} = \{x ∈ M | C ≠ 0 at x\} and U = U_{S} ∩ U_{C}. We note that U ⊂ U_{R}.

A semi-Riemannian manifold (M, g), n ≥ 3, is said to be pseudosymmetric if at every point of M the tensors R · R and Q(g, R) are linearly dependent. This is equivalent to

R · R = L_{R} Q(g, R)  (1)

on U_{R}, where L_{R} is some function on U_{R}. The class of pseudosymmetric manifolds is an extension of the class of semisymmetric manifolds ( R · R = 0). Some geometrical considerations show that (1) is a more natural curvature condition than the condition of semisymmetry. For a presentation of facts related to this statement and other conditions of pseudosymmetry type we refer to a recent review paper [1].

2. Main resuts

Let (M, g), n = dim M ≥ 4, be a semi-Riemannian manifold satisfying at every point the following curvature condition of pseudosymmetry type: the tensors R · C and Q(S, C) are linearly dependent. This is equivalent on the set U consisting of all points of M at which Q(S, C) ≠ 0 to

R · C = L Q(S, C) ,  (2)

where L is some function on U. In this paper, without loss of generality, we restrict our investigations to the set U_{L} ⊂ U defined by U_{L} = {x ∈ U | L ≠ 0 at x}.

Our aim is to investigate hypersurfaces in semi-Riemannian space of constant curvature N_{s}^{n+1}(c), n ≥ 4, with signature (s, n + 1 − s). Let M be a such hypersurface. We denote by g the metric tensor of M induced from the metric of the ambient space and by H the second fundamental tensor of (M, g). Let U_{H} be the subset of M consisting of all points at which the tensor H^{2} is not a linear combination of H and g. In a few earlier papers we considered hypersurfaces in semi-Euclidean spaces 𝔼_{s}^{n+1}, n ≥ 4, fulfilling (2). For instance, in [4] (see Theorem 4.1 and Theorem 4.2) it was shown that if M is a hypersurface in 𝔼_{s}^{n+1}, n ≥ 4, satisfying (2) then on U_{H} ∩U_{L} we have: rank (S − \frac{κ}{n−1} g) = 1 and R·C = Q(S, C) , i.e. (2) with L = 1. On the other hand we have

Proposition 1 ([3], Proposition 4.3). Let M be a hypersurface in N_{s}^{n+1}(c) satisfying R · C = Q(S, C). If U_{H} ⊂ M is nonempty then the ambient space must be semi-Euclidean.

Therefore we investigate hypersurfaces M in N_{s}^{n+1}(c) with nonzero sectional curvature c satisfying (2). The main result is the following

Theorem 2 ([3], Theorem 4.1). Let M be a hypersurface in N_{s}^{n+1}(c), c ≠ 0, satisfying (2). Then at every point x ∈ U_{H} ∩ U_{L} ⊂ M we have

\frac{κ}{n − 1} = \frac{κ̃}{n + 1} , S − \frac{κ}{n} g = β w ⊗ w , β ∈ , w ∈ T_{x}^{∗}M

and

L = \frac{1}{n − 1} , R · R = \frac{κ}{n(n − 1)} Q(g, R) , R · R = Q(S, R) − \frac{(n − 2)κ}{n(n − 1)} Q(g, C) .

From Theorem 5.1 of [2] it follows: (1) holds at a point x ∈ U_{H} if and only if at this point rank H = 2, i.e. the type number of M at this point is equal to 2. We have the following

Proposition 2 ([3], Proposition 5.1). Let M be a pseudosymmetric hypersurface in N_{s}^{n+1}(c), n ≥ 4.

(i) At every point x ∈ U_{H} ⊂ M we have

(a) rank (H^{2} − tr(H) H) = 1 , or (b) rank (H^{2} − tr(H) H) = 2 ,  (3)

(ii) If c ≠ 0 and at x ∈ U_{H} we have (3)(b) then (2) cannot be satisfied at this point. (iii) If c ≠ 0 and at a point x ∈ U_{H} we have (3)(a), i.e.

H_{ij}^{2} − tr(H)H_{ij} = \frac{1}{ρ} a_{i}a_{j} , ρ ∈

then the following relations are fulfilled at x: (2) with L = 1/(n − 1), \frac{κ}{n−1} = \frac{κ̃}{n+1} and

a^{k}a_{k} = 0 , a^{k} = g^{jk}a_{j} , S_{ij} − \frac{κ}{n} g_{ij} = \frac{ε}{ρ} a_{i}a_{j} ,

Basing on this result we construct two examples of hypersurfaces with type number two. The first one satisfies the equality R · C = \frac{1}{n−1} Q(S, C) and the second one does not.

References

[1] M. Belkhelfa, R. Deszcz, M. Głogowska, M. Hotloś, D. Kowalczyk, and L. Verstraelen, A review on pseudosymmetry type manifolds, Proc. Banach Center, in print.

[2] R. Deszcz, Pseudosymmetric hypersurfaces in spaces of constant curvature, Tensor, N.S. 58 (1997), 253–269.

[3] R. Deszcz and M. Hotloś, On hypersurfaces with type number two in space forms, to appear.

[4] R. Deszcz, M. Hotloś, and Z. S.entürk, Quasi-Einstein hypersurfaces in semi-Riemannian space forms, Colloq. Math. 89 (2001), 81–97.

Institute of Mathematics Wrocław University of Technology Wybrzeże Wyspiańskiego 27 50–370 Wrocław POLAND e-mail: hotlos@im.pwr.wroc.pl

COMBINATORICS OF THE FIRST NEIGHBOURHOOD OF THE DIAGONAL

ANDERS KOCK

Abstract

The consideration of the k’th neighbourhood of the diagonal of a manifold M

M_{(k)} ⊆ M × M

was initiated by Grothendieck to import notions from differential geometry into the realm of algebraic geometry. These notions were re-imported into differential geometry by Malgrange. Grothendieck and Malgrange utilzed the notion of ringed space (a space equipped with a structure sheaf of functions). The only points of the (underlying space of) M_{(k)} are the diagonal points (x, x) with x ∈ M. But it is worthwhile to describe mappings to and from M_{(k)} as if it consisted of “pairs of k-neighbour points (x, y)" (write x ∼_{k} y for such a pair; such x and y are “point proches" in the terminology of A. Weil). The introduction of topos theoretic methods has put this “synthetic" way of speaking onto a rigourous basis, and we shall freely use it.

A differential 1-form ω on M is thus defined to be a map ω : M_{(1)} → R, vanishing on the diagonal. So ω(x, y) makes sense whenever x ∼_{1} y; and ω(x, x) = 0 for all x ∈ M. Unravelling the definition of M_{(1)} in terms of its structure sheaf almost immediately reveals that such an ω is an element of the Kähler differentials Ω^{1}(M) = I/I^{2}. There is no linearity requirement on ω; and ω(x, y) = −ω(y, x) is automatic.

One can go on and define a k-form on M as an element of the k’th exterior power of Ω^{1}. This is the classical approach in algebraic geometry. But there is an alternative, more geometric/simplicial approach to the theory of differential forms, which we shall expound.

It is based on the consideration of the space

M_{[k]} ⊆ M^{k+1}

of “infinitesimal k-simplices". It is the “set" of k + 1-tuples (x_{0}, …, x_{k}) with x_{i} ∼_{1} x_{j} for all i, j = 0, …, k. We shall call the simplex degenerate if two of its vertices x_{i} and x_{j} are equal. Then the geometric/synthetic/combinatorial approach to differential forms is based on

Theorem There is a bijective correspondence between functions ω : M_{[k]} → R vanishing on degenarate simplices, and classical differential k-forms on M.

Note that there is no multilinearity or alternating requirement on ω. – Since the M_{[k]}’s jointly form a simplicial “set”, a differential k-form may be seen as a k-cochain, and there is therefore a formula for its coboundary; for instance, if ω is a 1-form, dω is the 2-form given by

dω(x, y, z) = ω(x, y) + ω(y, z) + ω(z, x).  (1)

It can be proved to correspond to the classical exterior derivative, under the correspondence of the theorem. But it generalizes in a more seamless way to differential forms with values in Lie groups G more general than R. For instance, (1) is replaced by

dω(x, y, z) = ω(x, y) · ω(y, z) · ω(z, x).  (2)

This aspect of the theory is well suited for a treatment of the theory of connections in principal G-bundles P or groupoids P P^{−1}. A connection in a bundle E → M is, in the synthetic theory, a law ∇ which to a pair of 1-neighbour points x ∼_{1} y in M associates a “parallel transport” map ∇_{yx} from the x-fibre of E to its y-fibre.

We shall describe the relationship between gauge-P valued forms, connections, and curvature, in synthetic/combinatorial terms. The curvature R_{∇} of ∇ is a 2-form (in the above sense) on M with values in a suitable group bundle on M, and is given by

R_{∇}(x, y, z) = ∇(x, y) ∘ ∇(y, z) ∘ ∇(z, x),

– We shall utilize these descriptions to provide an explicit construction of a connection in a principal G-bundle P → M, out of the data of a G-valued Čech-cocycle for the bundle P, and an R-valued partition of unity on M; this is based on the possibility of forming arbitrary affine combinations of the vertices of an infinitesimal simplex.

Department of Mathematical Science

University of Aarhus

DK 8000 Aarhus C

DENMARK

e-mail: kock@imf.au.dk

   http://home.imf.au.dk/kock/

ON FIBRATION WITH GRASSMANN MANIFOLDS AS FIBRES

JÚLIUS KORBAŠ

Abstract

The aim of this talk is to present some results on Z_{2}-cohomology properties of smooth fiber bundles and of Serre fibrations having complex or real Grassmann manifolds as their fibers.

We mainly concentrate on describing situations when the fiber inclusion (into the total space) induces an epimorphism in Z_{2}-cohomology. This is related to other topological questions.

The talk will be based on the paper On fibrations with Grassmannian fibers, Bull. Belgian Math. Soc. 8 (2001), 119–130; more recent results will also be included.

Department of Algebra Faculty of Mathematics and Physics Comenius University Mlynská dolina SK-842 48 Bratislava 4 SLOVAKIA e-mail: korbas@fmph.uniba.sk

TENSOR PRODUCT OF MODULES OVER NON–UNITAL ALGEBRAS AND LIE–RINEHART ALGEBRAS

JAN KUBARSKI

Abstract

In 1990–99 J. Huebschmann wrote a series of papers relating to Lie–Rinehart algebras. In

J. Huebschmann, Poisson cohomology and quantization, J. für die Reine und Angew.

Math. 408 (1990), 57–113

the base of the series – the author wrote [p. 70] ”Let A be an algebra over R, not necessarily with 1” and repeated this sentence in the context of commutative algebras on next pages. However, all the technical tools which were used, are appropriate in the case of unital algebras only. If we consider the typical situation where the base ring R is unital, then non-unitality of the R–algebra A means that there is no homomorphism of rings l : R → A such that l(r)a = ra = al(r). There are some simple anomalies in the theory of A–modules over non-unital R–algebra A which caused that the planned researches on Lie–Rinehart algebras for algebras not necessarily with 1 failed. Moreover, the construction of the Picard group for a ring A with a unit can not be adapted to the non–unital case. The reason is that there is a difference between projective modules in the category of non–unital and unital modules. The aim of this paper is to construct the notion of a tensor product of modules over non–unital algebras which does not possesses the anomalies and its applications for the Picard group of non–unital algebras and Lie–Rinehart algebras.

Institute of Mathematics

Technical University of Łódź

al. Politechniki 11, 90–924 Łódź

POLAND

e-mail: kubarski@ck-sg.p.lodz.pl

SPACES OF DIFFERENTIAL OPERATORS AS MODULES OVER THE LIE ALGEBRA OF VECTOR FIELDS

PIERRE LECOMTE

Abstract

The space D_{λ} of differential operators acting on the λ-densities over a smooth manifold M is filtered by the order of differentiation. The associated graded space is the space S(M) of smooth functions on T^{∗}M that are polynomial on the fibers. Both D_{λ} and S(M) are modules over the Lie algebra of vector fields V ect(M) of M and the projection map D_{λ} → S(M), called the principal symbol map, is equivariant. It is onto. Moreover, although as vector spaces, D_{λ} and S(M) are isomorphic, there is no equivariant bijection between them.

Viewing T^{∗}M as the phase space of some mechanical system, S(M) is then the Poisson algebra of the classical observables. A bijection from S(M) onto D_{λ} that preserves the principal symbol could then be interpreted as a quantification procedure. From the infinitesimal point of view, symmetries of the system lead to vector fields leaving that procedure equivariant.

It is known that there is no natural quantization, that is no quantization procedure that is equivariant under V ect(M). On the other hand, quantization procedures have been constructed on IR^{m} that are equivariant under the projective embedding sl_{m+1} of sl(m + 1, IR) and the conformal embedding of so(p + 1, q + 1), p + q = m. Moreover some uniqueness properties have been shown for these quantizations. In particular, the sl_{m+1}-equivariant quantization is unique. This has been used to study various questions about V ect(M)-modules of differential operators over arbitrary manifolds, the strategy being as follows: first study them over IR^{m}, filtering V ect(IR^{m}) by sl_{m+1} then glue the local informations collected on the various domains of chart by this means to get a global result. The power of the method comes from the fact that the filtering algebra is finite dimensional and simple, simplifying for instance cohomological considerations.

The above algebras of symmetries turn out to be maximal subalgebras of the Lie algebra of polynomial vector fields of IR^{m}. Because of that the family of these maximal subalgebras has been studied since then as well as the corresponding existence and uniqueness problem for the corresponding quantization procedures. The main result about the maximal subalgebras is that they coincide with the well known filtered algebras studied by Kobayashi and Nagano, that are related to geometries of order 2. Besides, algorithms have been found to decide wether or not the corresponding quantization procedures exist and are unique, using the ressources of the represntation theory of the semisimple Lie algebras. 3 The existence problem of quantization procedures is a particular case of the more general problem of classification of the spaces of differential operators as modules over Lie subalgebras of the algebra of vector fields. These questions involved some cohomological considerations that have also been in vestigated, leading to some nice universal cocycles.

On the other hand, going from vector space to curved manifold in order to get coordinate free expression of these quantization procedures also poses nice questions that one has started to study.

My goal would be to present a landscape of all that stuff, presenting the main results, the main methods and tools and the main contributors in the field.

Departement of Mathematics of the University of Liege Grande Traverse 12 (B37), B-4000 Liege (Sart Tilman) BELGIUM e-mail: plecomte@ulg.ac.be

http://www.forum.math.ulg.ac.be
http://www.ulg.ac.be/geothalg

SHORTLY ON NATURAL BUNDLES

WŁODZIMIERZ MIKULSKI

Abstract

In this year it is celebrated centenary of Polish mathematician Professor Stanisław Goła̧b at Jagellonian University, who formulated the definition of geometric objects and created ”Polish Shool” investigating these objects. On this occasion a short survey on so called natural bundles and bundle functors is presented. A new result is presented, too. Namely, a bundle functor on Mf_{m} × Mf of infinite order in the first factor is constructed

Introduction

The modern development of the differential geometry clarified that geometric objects (see [1]) are sections of fiber bundles over manifolds. (For example, the vector fields on a manifold M are sections of the tangent bundle T (M) of M, the 1-forms on M are sections of the cotangent bundle T^{∗}(M), etc.) Each type of geometric objects can be interpreted as a rule F transforming every m-dimensional manifold M into a fibered manifold F (M) → M over M and every local diffeomorphism f : M → N into a fibered manifold morphism F (f) : F (M) → F (N) over f. The geometric character of F is then expressed by the functoriality condition F (g ∘ f) = F (g) ∘ F (f). Hence the classical bundles of geometric objects are now studied in the form of the so called natural bundles on the category Mf_{m} of all m-dimensional manifolds and their local diffeomorphism.

All manifolds are assumed to be finite dimensional, paracompact, second countable, without boundaries and smooth, i.e. of class C^{∞}. Maps between manifolds are assumed to be smooth.

I. Natural bundles over m-manifolds

Let us present a definition of natural bundles over m-manifolds.

Definition 1. (Nijenhuis, [22]) A natural bundle over m-manifolds is a covariant functor F : Mf_{m} → FM from the category Mf_{m} of m-dimensional manifolds and local diffeomorphisms into the category FM of fibered manifolds and fibered maps satisfying the following conditions:

(i) (Prolongation) For every m-manifold M F (M) is a fibered manifold over M with projection p_{M} : F (M) → M and for every local diffeomorphism f : M → N of two m-manifolds F (f) : F (M) → F (N) is a fibered map covering f;

(ii) (Locality) If i : U → M is an inclusion of an open submanifold, then F (U) = p_{M}^{−1}(U) and F (i) is the inclusion of p_{M}^{−1}(U) into F (M).

(iii) (Regularity) F transform smoothly parametrized systems of Mf_{m}-morphisms into smoothly parametrized systems of FM-morphisms.

Simple examples of natural bundles are following.

Example 1. The tangent bundle. T (M) denotes the tangent bundle of M and T (f) : T (M) → T (N) is the tangent map of f : M → N. The functor T : Mf_{m} → FM is a natural bundle over m-manifolds.

3Key words: natural bundles, gauge natural bundles, bundle functors, jets, Weil algebras, Weil bundles

AMS classification: 53 A 55, 58 A 05, 58 A 20

Example 2. The cotangent bundle. T^{∗}(M) denotes the cotangent bundle of M, and for a local diffeomorphism f : M → N and a point x ∈ M let (T_{x}(f))^{∗} : T_{f(x)}^{∗}(N) → T_{x}^{∗}(M) be the transposed mapping of the differential T_{x}(f) : T_{x}(M) → T_{f(x)}(N) of f at x . Then T_{x}(f) is a linear isomorphism, and we define T_{x}^{∗}(f) = ((T_{x}(f))^{∗})^{−1} : T_{x}^{∗}(M) → T_{f(x)}^{∗}(N) and the induced mapping T^{∗}(f) : T^{∗}(M) → T^{∗}(N) by (T^{∗}(f))_{|T_{x}^{∗}(M)} = T_{x}^{∗}(f) . The functor T^{∗} : Mf_{m} → VB is a natural bundle over m-manifolds.

Example 3. The bundle of frames of order r. Let m and r be natural numbers. We denote by G_{m}^{r} the differential group of order r in dimension m, it means

G_{m}^{r} = { j_{0}^{r}(ξ) | ξ : U → R^{m}, U ⊂ R^{m} is open, ξ is a local diffeom., ξ(0) = 0 } ,

where j_{0}^{r}(ξ) is the r-jet of ξ at 0. G_{m}^{r} is a Lie group with respect to r-jets multiplication j_{0}^{r}(ξ_{1}) · j_{0}^{r}(ξ_{2}) = j_{0}^{r}(ξ_{1} ∘ ξ_{2}).

Let M be an m-manifold. We denote by P^{r}(M) the r-th order frame bundle of M over M. It means

P^{r}(M) = { j_{0}^{r}(γ) | γ : U → M, U ⊂ R^{m} is open, γ is a local diffeomorphism } .

The group G_{m}^{r} acts on P^{r}(M) on the right by the formula j_{0}^{r}(γ) · j_{0}^{r}(ξ) = j_{0}^{r}(γ ∘ ξ). P^{r}(M) is a principal fibre bundle with the structure group G_{m}^{r}. For a local diffeomorphism f : M → N of two m-manifolds the induced mapping P^{r}(f) : P^{r}(M) → P^{r}(N) is given by P^{r}(f)(j_{0}^{r}(γ)) = j_{0}^{r}(f ∘ γ). The functor P^{r} : Mf_{m} → FM is a natural bundle over m-manifolds.

The most general example of natural bundles is following.

Example 4. The associated bundle. Let S be an G_{m}^{r}-space, it means that S is a manifold and G_{m}^{r} acts on S on the left by an action α. For an m-manifold M we put

E^{S}(M) = P^{r}(M)[S, α] ,

where P^{r}(M)[S, α] is the associated bundle with P^{r}(M) and standard fibre S. We recall that P^{r}(M)[S, α] is the set of all orbits in P^{r}(M) × S with respect to the action of G_{m}^{r} given by (j_{0}^{r}(γ), y) · j_{0}^{r}(ξ) = (j_{0}^{r}(γ ∘ ξ), j_{0}^{r}(ξ^{−1}) · y) . We denote by < j_{0}^{r}(γ), y > the orbit of (j_{0}^{r}(γ), y) and we define π : E^{S}(M) → M , π(< j_{0}^{r}(γ), y >) = γ(0) .

For a local diffeomorphism f : M → M of two m-manifolds we define E^{S}(f) : E^{S}(M) → E^{S}(N) by

E^{S}(f)(< j_{0}^{r}(γ), y >) =< j_{0}^{r}(f ∘ γ), y > .

The functor E^{S} : Mf_{m} → FM is a natural bundle over m-manifolds.

Definition 2. Let F be a natural bundle over m-manifolds. We say that F is of finite order if there is a number r such that from j_{x}^{r}(f) = j_{x}^{r}(g) it follows F_{x}(f) = F_{x}(g) for every local diffeomorphisms f, g : M → N between m-manifolds and every point x ∈ M, where (obviously) F_{x}(f) : F_{x}(M) → F_{f(x)}(N) is the restriction and corestriction of F (f) : F (M) → F (N) to the fibres F_{x}(M) and F_{f(x)}(N) of F (M) and F (N) over x ∈ M and f(x) ∈ N. The smallest number r satisfying the above property is called the order of F.

The tangent bundle T is of order 1, the cotangent bundle T^{∗} is of order 1, the bundle P^{r} of frames of order r is of order r and the associated (with P^{r}) bundle E^{S} is at most of order r.

Theorem 1. (Palais and Terng, [23]) Every natural bundle F over m-manifolds has a finite order. The order of F is less than or equal to 2^{f} + 1, where f = dim(F_{0}(R^{m})).

In [16], Masuda investigated homomorphisms of Lie algebras of vector fields. From the results of Masuda one can very easily deduce the above order theorem (even with a better estimation). Namely, from the results of [16] it follows that the order of a natural bundle F over m-manifolds is less than or equal 2(f^{2} + f) + 1 , see [17]. The paper of Masuda was published one year ealier than the paper of Palais and Terng. However Masuda has not formulated the order theorem explicitly.

Theorem 2. (Epstein and Thurston, [5]) If F : Mf_{m} →FM is a covariant functor from Definition 1 satisfying the conditions (i) and (ii), then F satisfies also (iii) and F has a finite order less than or equal to 2f + 1, where f = dim(F_{0}(R^{m})). If m = 1, the estimation of the order is sharp.

Theorem 3. (Zajtz, [26]) Let F be a natural bundle over m-dimensional manifolds, m ≥ 2, with a standard fibre of dimension f. Then the order r of F satisfies the inequality

r ≤ max(\frac{f}{m − 1}, \frac{f}{m} + 1) .

This estimation of the order is sharp.

Let F be a natural bundle over m-manifolds. According to Theorem 1 F has a finite order r. Let S_{F} = F_{0}(R^{m}) be the standard fibre of F. We can define an action α_{F} on the left of the r-th order differential group G_{m}^{r} on S_{F} by the formula

j_{0}^{r}(ξ) · y = F (ξ)(y) ,

where y ∈ S_{F} and j_{0}^{r}(ξ) ∈ G_{m}^{r} . This action is well-defined because of the order argument and it is smooth because of the regularity condition (iii) of Definition 1.

The above action is called the standard action of G_{m}^{r} on the standard fibre S_{F} of F. Now we can consider the associated bundle E^{S_{F}} : Mf_{m} → FM.

Theorem 4. (Palais and Terng, [23]) Every natural bundle F over m-manifolds is equivalent with the associated bundle E^{S_{F}}, where S_{F} = F_{0}(R^{m}) is the standard fibre of F with the standard action of G_{m}^{r} on S_{F}, where r is the order of F.

Proof. (see [6]) A canonical diffeomorphism I_{M} : E^{S_{F}} (M) ⟶ F (M) is given by I_{M}(< j_{0}^{r}(γ), y >) = F (γ)(y) .

In particular case when F (M) is a principal bundle Theorem 4 was proved independently by Krupka, [13].

II. Bundle functors on manifolds

Some ”natural bundles” (as T) can be defined on the whole category Mf of all manifolds and maps.

Definition 3. A bundle functor on Mf is a covariant functor F : Mf → FM satisfying the following conditions (i)—(iii) of the definition of natural bundles with Mf instead of Mf_{m}.

We have the following examples of bundle functors on Mf:

Example 5. The tangent bundle T : Mf →FM is a bundle functor on Mf.

Example 6. The bundle T_{p}^{r} of p^{r}-velocities. The r-tangent bundle T^{r}. Let M be a manifold. Let T_{p}^{r}(M) = J_{0}^{r}(R^{p}, M) be the bundle of r-jets at 0 ∈ R^{p} of smooth functions R^{p} → M. T_{p}^{r}(M) is called the bundle of p^{r}-velocities over M. Every mapping f : M → N of two manifolds is extended to a fibre bundle mapping T_{p}^{r}(f) : T_{p}^{r}(M) → T_{p}^{r}(N), j_{0}^{r}(γ) → j_{0}^{r}(f ∘ γ). The functor T_{k}^{r} : Mf → FM is a bundle functor.

For p = 1 the functor T^{r} = T_{1}^{r} is called the r-tangent bundle functor.

Example 7. The r-th order vector tangent bundle T^{(r)}. Let M be a manifold. Let T^{(r)}(M) = (T^{r∗}(M))^{∗} be the dual vector bundle, where T^{r∗}(M) = J^{r}(M, R)_{0}. Vector bundle T^{(r)}(M) is called the r-th order vector tangent bundle of M. Every smooth mapping f : M → N of two manifolds induces vector bundle mapping T^{(r)}(f) : T^{(r)}(M) → T^{(r)}(N) as follows. We denote the fibres of T^{(r)}(M) and T^{r∗}(M) over a point x ∈ M by T_{x}^{(r)}(M) and T_{x}^{r∗}(M) respectively. For any x ∈ M we have the linear mapping f_{|x}^{r∗} : T_{f(x)}^{r∗}(N) → T_{x}^{r∗}(M) defined by f_{|x}^{r∗}(j_{f(x)}^{r}(γ)) = j_{x}^{r}(γ ∘ f). We define T^{(r)}(f) : T^{(r)}(M) → T^{(r)}(N) by

the condition that the restriction T^{(r)}(f)_{|T_{x}^{(r)}(M)} : T_{x}^{(r)}(M) → T_{f(x)}^{(r)}(N) is the transposed

mapping to f_{|x}^{r∗} . The functor T^{(r)} : Mf → VB is a vector bundle functor on Mf.

To present a next example we need a preparation.

Let L^{r} = {L_{m,n}^{r}}_{m,n=0,1,2,...}, where

L_{m,n}^{r} = J_{0}^{r}(R^{m}, R^{n})_{0}

is the space of r-jets at 0 ∈ R^{m} of maps R^{m} → R^{n} sending 0 into 0.

Let S = {S_{0}, S_{1}, ...} be a system of manifolds. An action α of L^{r} on S is a system of smooth maps α_{m,n} : L_{m,n}^{r} × S_{m} → S_{n} satisfying

α_{m,p}(B ∘ A, s) = α_{n,p}(B, α_{m,n}(A, s))

for every A ∈ L_{m,n}^{r}, B ∈ L_{n,p}^{r} and s ∈ S_{m}.

Example 8. The associated bundle. Consider a system of manifolds S = {S_{0}, S_{1}, ...} and an action α of L^{r} on S. We shall construct a bundle functor E^{α} determined by this action. The restriction α_{m} of the maps α_{m,m} to invertible jets form actions of the jet group G_{m}^{r} on S_{m}. For every m-manifold M we put

E^{α}(M) = P^{r}(M)[S_{m}, α_{m}] .

For every map f : M → N we define E^{α}(f) : E^{α}(M) → E^{α}(N) by

E^{α}(f)(< u, s >) =< v, α_{m,n}(v^{−1} ∘ A ∘ u, s) > ,

where m = dim(M), n = dim(n), u ∈ P_{x}^{r}(M), A = j_{x}^{r}(f), s ∈ S_{m} and v ∈ P_{f(x)}^{r}(N) is an arbitrary element. One can easily seen that this is a corect definition and the correspondence E^{α} : Mf → FM is a bundle functor on Mf.

Definition 4. Let r be a non-negative integer or infinity. Let F be a bundle functor on Mf. We say that F is of order less or equal to r if from j_{x}^{r}(f) = j_{x}^{r}(g) it follows F_{x}(f) = F_{x}(g) for every mappings f, g : M → N and every point x ∈ M. The smallest number r satisfying the above property is called the order of F.

Theorem 5. (Mikulski, [19]) Every bundle functor F on Mf has locally a finite order. More precisely, for any manifolds M, N, any point x ∈ M and any mappings f, g : M → N the condition j_{x}^{r(dim(M)+1)}(f) = j_{x}^{r(dim(M)+1)}(g) implies F_{x}(f) = F_{x}(g), where r(m) denotes the order of the natural bundle obtained by the restriction of F to the category of m-dimensional manifolds and their local diffeomorphisms.

An open problem. Whether the condition j_{x}^{r(dim(M))}(f) = j_{x}^{r(dim(M))}(g) implies F_{x}(f) = F_{x}(g) for any manifolds M, N, any point x ∈ M and any mappings f, g : M → N, i.e. is the estimation in Theorem 5 sharp.

Example 9. (Mikulski, [18]) A bundle functor on Mf of infinite order. We recall that T^{(r)}(M) denotes the r-th order vector tangent bundle (see Example 7). Let d_{r} = dim(T_{0}^{(r)}(R^{r})). We set

F (M) = ⨁_{k=1}^{+∞} (⋀^{d_{k}} T^{(k)}(M)) .

For every manifold M, F (M) is a finite dimensional manifold because for k > dim(M) the bundle ⋀^{d_{k}} T^{(k)}(M) is zero-bundle. Hence the direct sum in the definition of F (M) is in reality a finite sum. For a mapping f : M → N the induced mapping F (f) : F (M) → F (N) is defined in the natural way from ⋀^{d_{k}} T^{(k)}(f).

The bundle functor F is of infinite order because its restriction to the category of kdimensional manifolds is at least of order k.

Theorem 6. (Kolář and Slovák, [12]) If F is a covariant functor from Definition 3 satisfying the conditions (i) and (ii), then F satisfies also (iii).

Let F be a bundle functor on Mf of finite order r. Put S^{F} = {S_{0}^{F} , S_{1}^{F} , ..., }, where S_{m}^{F} = F_{0}(R^{m}) for m = 0, 1, ... are the standard fibers of F. We can define an action α^{F} = {α_{m,n}^{F}} of L^{r} on S^{F} by the formula

j_{0}^{r}(h) · s = F (h)(s) ,

where s ∈ S_{m}^{F} and j_{0}^{r}(h) ∈ J_{0}^{r}(R^{m}, R^{n})_{0}. This action is well defined because of the order argument, and it is smooth because of the regularity condition (iii).

The above action is called the standard action of L^{r} on the standard fibres S^{F} of F. Now we can consider the associated bundle E^{α^{F}} : Mf → FM.

Theorem 7. (Janyška, [7]) Every bundle functor F on Mf of finite order r is equivalent with the associated bundle E^{α^{F}}.

Proof. A canonical diffeomorphism I_{M} : E^{α^{F}} (M) ⟶ F (M) is given by I_{M}(< j_{0}^{r}(γ), s >) = F (γ)(s) .

III. Product preserving bundle functors on manifolds

Many bundle functors on Mf are product preserving.

Definition 5. Let F : Mf → FM be a bundle functor. For two manifolds M_{1}, M_{2} we denote the standard projection onto i-th factor by pr_{i} : M_{1} × M_{2} → M_{i}, where i = 1, 2. F is called product preserving if the mapping

(F (pr_{1}), F (pr_{2})) : F (M_{1} × M_{2}) → F (M_{1}) × F (M_{2})

is a diffeomorphism for all manifolds M_{1}, M_{2}.

Roughly speaking, F is product preserving if F (M_{1} × M_{2}) = F (M_{1}) × F (M_{2}) for any manifolds M_{1} and M_{2}.

Example 10. The tangent bundle T : Mf → FM is a product preserving bundle functor.

Example 11. The p^{r}-velocities bundle T_{p}^{r} : Mf → FM is a product preserving bundle functor.

To present a next example of product preserving bundle functors on Mf we need a preparation.

Definition 6. A finite dimensional real commutative associative unital algebra A is called a Weil algebra if it is of the form A = R · 1 ⊕ N, where N is a nilpotent ideal.

Remark 1. Let C_{0}^{∞}(R^{p}) be the algebra of all germs at 0 of maps R^{p} → R. It is a local algebra and the ideal m of all germs R^{p} → R vanishing at 0 is its maximal ideal. Let \underline{A} ⊂ C_{0}^{∞}(R^{p}) be an ideal such that m ⊃ \underline{A} ⊃ m^{κ+1} for some natural number κ, where m^{κ+1} is the algebraic κ + 1 power of m. The factor algebra A = C_{0}^{∞}(R^{p})/\underline{A} is a Weil algebra. Any Weil algebra is isomoprhic to some A = C_{0}^{∞}(R^{p})/\underline{A}.

Example 12. (Weil, [25]) The Weil bundle T^{A} = Hom(., A) of near A-points. Let A be a Weil algebra. Let M be a manifold. Let C_{x}^{∞}(M) be the algebra of all germs at x ∈ M of mappings M → R. An unital algebra homomorphism v : C_{x}^{∞}(M) → A is called a near A-point over x ∈ M. The space of all near A-points over x will be denoted by Hom(C_{x}^{∞}(M), A). Let

T^{A}(M) = ⋃_{x∈M} Hom(C_{x}^{∞}(M), A) .

T^{A}(M) is a smooth fiber bundle over M with the obvious projection. Every map f : M → N of two manifolds is extended to a fibre bundle map T^{A}(f) : T^{A}(M) → T^{A}(N) over f by

T^{A}(f)(v)(germ_{f(x)}(g)) = v(germ_{x}(g ∘ f)) ,

x ∈ M, v ∈ Hom(C_{x}^{∞}(M), A), g : N → R. The functor T^{A} : Mf → FM is a product preserving bundle functor. It is of finite order less or equal to the nilpotency order of A.

Let us denote the addition and the multiplication on R by +, · : R^{2} → R and for λ ∈ R denote m_{λ} : R → R the scalar multiplication by λ. If we apply a product preserving bundle functor F we obtain F (+) : F (R) × F (R) = F (R × R) → F (R), F (·) : F (R) × F (R) → F (R) and F (m_{λ}) : F (R) → F (R). Now, we can formulate the key lemma.

Lemma 1 (Kainz and Michor, [8], Eck, [3], Luciano, [15]) If F is a product preserving bundle functor on Mf, then A^{F} = F (R) is a Weil algebra with operations F (+), F (·), F (m_{λ}), zero F (0) and unit F (1), and F_{0}(R) is its nilpotent ideal.

Proof. Let r be the order of the restriction of F to the category Mf_{1}. Then r is finite because of Theorem 1. The map f : R → R, f(x) = x + x^{r+1}, satisfies j_{0}^{r}(f) = j_{0}^{r} (id_{R}). Then F_{0}(f) = id_{F_{0}(R)}. Hence a + a^{r+1} = F_{0}(f)(a) = a, i.e. a^{r+1} = 0 for any a ∈ F_{0}(R).

We call A^{F} the Weil algebra of F.

Then we have Weil bundle T^{A^{F}} of near A^{F} -points.

Theorem 8. (Kainz and Michor, [8], Eck, [3], Luciano, [15]) Every product preserving bundle functor F on Mf is equivalent to the Weil bundle T^{A^{F}} of near A^{F} point, where A^{F} = F (R) is the Weil algebra of F.

Proof. A canonical diffeomorphism I_{M} : F (M) → T^{A^{F}} (M) can be constructed as follows. If v ∈ F_{x}(M) we have an algebra homomorphism ṽ : C_{x}^{∞}(M) → A^{F} by ṽ(germ_{x}(g)) = F (g)(v) for any g : M → R, and we put I_{M}(v) = ṽ.

Very interesting is also the following characterization:

Theorem 9. (Kolář and Slovák, [12]) A bundle functor F on Mf is product preserving if and only if card(F (R^{0})) = 1 and dim(F (R^{n})) = n · dim(F (R)) for any n = 0, 1, 2, ....

IV. Bundle functors on fibered manifolds and on some other local categories over manifolds

Some importrant ”bundle functors” are defined on the category FM_{m,n} of fibered manifolds with m-dimensional basis and n-dimensional fibers and their local fibered diffeomorphisms. Some of them are defined on a larger category FM_{m} of fibered manifolds with m-dimensional bases and fibered maps covering local diffeomorphisms. Some are defined on the whole category FM of all fibered manifolds and fibered maps. The definitions of such bundle functors are obvious modifications of the definition of natural bundles or bundle functors. Similarly we define the order of such bundle functors.

Example 13. The r-jet prolongation bundle functor. For an FM_{m}-object Y → M we set

J^{r}(Y) = { j_{x}^{r}(σ) | σ is a (locally defined) section of Y , x ∈ M}

Then J^{r}(Y) is a fibered manifold over Y with respect to the target projection. J^{r}(Y) is called the r-jet prolongation of Y. For an FM_{m}-morphism f : Y → Z covering a local diffeomorphism \underline{f} : M → N we have a fibered map J^{r}(f) : J^{r}(Y) → J^{r}(Z) covering f by the formula J^{r}(f)(j_{x}^{r}(σ)) = j_{\underline{f}(x)}^{r}(f ∘ σ ∘ \underline{f}^{−1}) . Functor J^{r} : FM_{m} → FM is a bundle functor on FM_{m}.

Let r, s, q be natural numbers such that q ≥ r ≤ s. Let Y → M and Z → N be fibered manifolds. Two fibered maps f, g : Y → Z with base maps \underline{f}, \underline{g} : M → N determine the same (r, s, q)-jet j_{y}^{r,s,q}(f) = j_{y}^{r,s,q}(g) at y ∈ Y_{x}, x ∈ M, if

j_{y}^{r}(f) = j_{y}^{r}(g) , j_{y}^{s}(f|Y_{x}) = j_{y}^{s}(g|Y_{x}) and j_{x}^{q}(\underline{f}) = j_{x}^{q}(\underline{g}) .

The space of all (r, s, q)-jets of Y into Z is denoted by J^{r,s,q}(Y, Z), see [10].

Example 14. The bundle of (k, l)^{r,s,q}-velocities. Let r, s, q, k, l be natural numbers with q ≥ r ≤ s. For any fibered manifold Y we define T_{k,l}^{r,s,q}(Y) = J_{0}^{r,s,q}(R^{k,l}, Y). It is a fiber bundle over Y with respect to the target projection. If f : Y → Z is a fibered map we have fibered map T_{k,l}^{r,s,q}(f) : T_{k,l}^{r,s,q}(Y) → T_{k,l}^{r,s,q}(Z) covering f by T_{k,l}^{r,s,q}(f)(j_{0}^{r,s,q}(γ)) = j_{0}^{r,s,q}(f ∘γ). The correspondence T_{k,l}^{r,s,q} : FM → FM is a bundle functor on FM.

For bundle functors on FM_{m,n} we have a similar characterization as in Theorem 4 (we construct the associated bundle in similar way as for m-manifolds by using FM_{m,n}-maps instead of FMf_{m}-maps). By Example 9, one can construct bundle functors on FM_{m} (or FM) of infinite order. Every bundle functor of FM_{m} is of locally finite order. For finite order bundle functors on FM_{m} (or FM) one can obtain similar characterization as in Theorem 7. In particular we have the following order theorem similar to Theorems 2, 3 and 5.

Theorem 10. (Slovák, [24]) (i) Let F : FM_{m,n} →FM be a functor from the definition of bundle functors satisfying conditions (i) and (ii) of the definition and let m ≥ 1 and n ≥ 0. Then F satisfies the regularity condition (iii) and it is of finite order r ≤ 2f + 1, where f is the dimension of the standad fiber S_{F} = F_{0}(R^{m,n}) of F. If moreover m > 1, n = 0, then

r ≤ max(\frac{f}{m − 1}, \frac{f}{m} + 1) ,

and if m > 1 and n > 1, then

r ≤ max(\frac{f}{m − 1}, \frac{f}{m} + 1, \frac{f}{n − 1}, \frac{f}{n} + 1) .

All these estimations are sharp.

(ii) Every bundle functor F on FM_{m} has locally a finite order. More precisely, for any fibered FM_{m,n}-manifold Y, any FM_{m}-object Z, any point y ∈ Y and any FM_{m}-maps f, g : Y → Z the condition j_{y}^{r(n+1)}(f) = j_{y}^{r(n+1)}(g) implies F_{y}(f) = F_{y}(g), where r(n) denotes the order of the bundle functor obtained by the restriction of F to the category FM_{m,n}.

Some bundle functors F (as T_{k,l}^{r,s,q}) on FM are product preserving, i.e. F (Y_{1} × Y_{2}) = F (Y_{1}) × F (Y_{2}). We have the following characterization of such functors.

Theorem 11. (Mikulski, [20]) (i) Every algebra homomorphism μ : A → B of Weil algebras determines (explicitely) a product preserving bundle functor T^{μ} : FM →FM. (ii) Conversely, any product preserving bundle functor F : FM →FM determines (explicitely) an algebra homomorphism μ^{F} : A^{F} → B^{F} of Weil algebras, and F ˜=T^{μ^{F}}.

A simple proof of Theorem 11 one can find also in [2].

Some bundle functors F (as J^{r}) on FM_{m} are fiber product preserving, i.e. F (Y_{1} ×_{M} Y_{2}) = F (Y_{1}) ×_{M} F (Y_{2}). We have the following characterization of such functors.

Theorem 11. (Kolář and Mikulski, [11]) (i) Every fiber product preserving bundle functor F : FM_{m} →FM is of finite order. (ii) Every triple (A, H, t), where A is a Weil algebra of nipotency order less or equal to a finite number r, H is a group homomorphism from the differential group G_{m}^{r} into the group Aut(A) of all automorphisms of A and t is a G_{m}^{r}-invariant algebra homomorphism from the Weil algebra D_{m}^{r} = J_{0}^{r}(R^{m}, R) into A, determines (explicitely) a fiber product preserving bundle functor T^{(A,H,t)} : FM_{m} →FM. (iii) Conversely, every fiber product preserving bundle functor F : FM_{m} →FM of order r determines (explicitely) a triple (A^{F} , H^{F} , t^{F}), where A^{F} is a Weil algebra of nipotency order less or equal to a finite number r, H^{F} is a group homomorphism from the differential group G_{m}^{r} into the group Aut(A^{F}) of all automorphisms of A^{F} and t^{F} is a G_{m}^{r}-invariant algebra homomorphism from the Weil algebra D_{m}^{r} = J_{0}^{r}(R^{m}, R) into A^{F}, and F ˜=T^{(A^{F} ,H^{F} ,t^{F})}.

Remark 2. A fiber product preserving bundle functors F : FM_{m} → FM such that J^{r}(Y) ⊂ F (Y) ⊂ J̃^{r}(Y) (the non holonomic r-jet prolongation of Y) is called of jet type. In [9], Kolář described completely all such bundle functors of jet type.

Remark 3. Roughly speaking, a local category over manifolds is a category C, whose objects can be interpreted as manifolds with ”structures” and morphisms as ”maps preserving structures”, such that ”open subsets of C-objects” are C-object and ”restrictions of C-morphisms to open subsets” are C-morphisms. For example, Mf_{m}, Mf, FM_{m,n}, FM_{m} and FM are local categories over manifolds. (For example FM is local because an open subset of a fibered manifold is again a fibered manifold and the restriction of a fibered map to an open subset is again a fibered map). In contrast, the category VB of vector bundles and their maps is not local (not every open subset U ⊂ E of a vector bundle E is a vector bundle). The concept of bundle functors F : C → FM on an arbitrary local category C over manifolds is presented in [10]. Roughly speaking a bundle functor on a local category C over manifolds is a covariant functor F : C → FM satisfying the conditions (i)—(iii) of Definition 1 (with C instad of Mf_{m}). In [10], under some assumptions on C the authors proved that condition (iii) for F : C → FM is a consequence of functoriality of F and conditions (i) and (ii) for F (i.e. a theorem similar to the first part of Theorem 2), and under some assumptions on C the authors presented a characterization of bundle functors F : C → FM of finite order (similar to the one given in Theorem 7). The mentioned above assumptions are satisfied for C = Mf_{m}, Mf, FM, FM_{m,n}, FM_{m}.

The category C = SympMf_{2m} of symplectic 2m-manifolds and their local symplectic diffeomorphisms is a local category over manifolds. The category C = V olMf_{m} of mmanifolds with locally integrable volume forms and their local diffeomorphisms preserving volume forms is local, too.

Theorem 12. (Lubaś and Zajtz, [14]) (i) Let F : SympMf_{2m} →FM be a covariant functor satisfying (i), (ii) and some stronger than (iii) regularity condition. If m ≥ 2, then F is of finite order

r ≤ \frac{f}{m} ,

where f = dim(F_{0}(R^{2m}) and R^{2m} is with the standard symplectic structure. The estimation is sharp.

(ii) Let F : V olMf_{m} →FM be a covariant functor satisfying (i), (ii) and some stronger than (iii) regularity condition. If m ≥ 2, then F is of finite order

r ≤ \frac{f}{m − 1} ,

where f = dim(F_{0}(R^{m}) and R^{m} is with the standard volume form. The estimation is sharp.

V. Bundle functors on some not local categories over manifolds

Fix a Lie group G and consider the category PB_{m}(G), whose objects are principal Gbundles over m-manifolds and whose morphisms are the morphisms of principal G-bundles f : P → \overline{P} with the base map lying in Mf_{m} and with homomorphisms of G equal to id_{G}. Since PB_{m}(G) is not local (not every open subset in a principal G-bundle P is again a principal G-bundle), there is a problem with the locality condition (ii) of bundle functors. To introduce the locality condition (ii) we must modify the prolongation condition (i).

Definition 7. (Eck, [4]) A gauge natural bundle on PB_{m}(G) is a covariant functor F : PB_{m}(G) → FM satisfying the following conditions:

(i) (Prolongation) For every PB_{m}(G)-object P → M F (P) is a fibered manifold over M (not over P) with projection p_{P} : F (P) → M and for every PB_{m}(G)-map f : P → \overline{P} covering \underline{f} : M → \overline{M} F (f) : F (P) → F (\overline{P}) is a fibered map covering \underline{f};

(ii) (Locality) If i : P |U → P is an inclusion, where U ⊂ M is open, then F (P |U) = p_{P}^{−1}(U) and F (i) is the inclusion of p_{P}^{−1}(U) into F (P).

(iii) (Regularity) F transform smoothly parametrized systems of PB_{m}(G)-morphisms into smoothly parametrized systems of FM-morphisms.

Example 15. The bundle of connection. Let P → M be an PB_{m}(G)-object with the right action r : P × G → P. We shall also denote by r the canonical right action r : J^{1}(P) × G → J^{1}(P) given by r^{g}(j_{x}^{1}(s)) = j_{x}^{1}(r^{g} ∘ s) for all g ∈ G and j_{x}^{1}(s) ∈ J^{1}(P). Then

Q(P) = J^{1}(P)/G ,

where J^{1}(P)/G is the set of orbits of the action r, is a bundle over M with projection [j_{x}^{1}(s)] → x. It is called the bundle of principal connections on P. (The smooth sections M → Q(P) are in bijection with principal connections on P.) Every PB_{m}(G)-map f : P → \overline{P} covering \underline{f} : M → \overline{M} induces a fibered map Q(f) : Q(P) → Q(\overline{P}) over \underline{f} by

Q(f)([j_{x}^{1}(s)]) = [j_{\underline{f}(x)}^{1}(f ∘ s ∘ \underline{f}^{−1})] .

The correspondence Q : PB_{m}(G) → FM is a gauge natural bundle.

If two PB_{m}(G)-maps f, g : P → \overline{P} satisfy j_{y}^{r}(f) = j_{y}^{r}(g) at a point x ∈ P_{x} of the fiber of P over x ∈ M, then the fact that the raight translations of principal bundles are diffeomorphisms implies j_{z}^{r}(f) = j_{z}^{r}(g) for every z ∈ P_{x}. In this case we write j_{x}^{r}(f) = j_{x}^{r}(g).

Definition 8. (Eck, [4]) Let F be a gauge natural bundle on PB_{m}(G). We say that F is of finite order if there is a number r such that from j_{x}^{r}(f) = j_{x}^{r}(g) it follows F_{x}(f) = F_{x}(g) for every PB_{m}(G)-maps f, g : P → \overline{P} covering \underline{f} : M → \overline{M} and every point x ∈ M. The smallest number r satisfying the above property is called the order of F.

We have the following order theorem similar to Theorems 2 and 3. This theorem was firstly proved by Eck, [4], but without a sharp estimation.

Theorem 13. (Eck, [4], Kolář, Michor, Slovák, [10]) Let F : PB_{m}(G) →FM be a functor from Definition 7 satisfying condition (i) and (ii), m ≥ 1. Then F satisfies condition (iii) and it is of finite order r ≤ 2f + 1, where f is the dimension of the standad fiber S_{F} = F_{0}(R^{m} × G) of F. If moreover m > 1, then

r ≤ max(\frac{f}{m − 1}, \frac{f}{m} + 1) .

All these estimations are sharp.

Theorem 13 gives a complete characterization of gauge natural bundles on PB_{m}(G) similar to Theorem 4 (we construct the associated bundle in similar way as for m-manifolds by using PB_{m}(G)-maps instead of Mf_{m}-maps). This characterization was obtained by Eck, [4].

Another not local category over manifolds is the category VB of all vector bundles and their vector bundle maps. Gauge bundle functors on VB we define similarly as gauge natural bundles on PB_{m}(G). We have the following complete characterization of product preserving gauge bundle functors on VB.

Theorem 14. (Mikulski, [21]) (i) Every pair (A, V) of a Weil algebra A and of an A-module V with dim_{R}(V) < ∞ determines (explicitely) a product preserving gauge bundle functor T^{(A,V)} : VB →FM. (ii) Conversely, any product preserving gauge bundle functor F : VB →FM determines (explicitely) a pair (A^{F} , V^{F}) of a Weil algebra A^{F} and of an A^{F} -module V^{F} with dim_{R}(V^{F}) < ∞, and F ˜=T^{(A^{F} ,V^{F})}.

Remark 4. Another not local category over manifolds is the product Mf_{m} × Mf of categories Mf_{m} and Mf. The objects of Mf_{m} × Mf are pairs (M, N), where M is an m-manifold and N is a manifold. The morphisms (M, N) → (\overline{M}, \overline{N}) are the pairs (φ, f), where φ : M → \overline{M} is a local diffeomorphism between m-manifolds and f : N → \overline{N} is a map. An example of a bundle functor F on Mf_{m} × Mf is the bundle functor J^{r}(M, N) of holonomic r-jets. Every pair of a local diffeomorphism φ : M → \overline{M} and a map f : N → \overline{N} induces fibered map J^{r}(φ, f) : J^{r}(M, N) → J^{r}(\overline{M}, \overline{N}) over φ × f by J^{r}(φ, f)(j_{x}^{r}(g)) = j_{φ(x)}^{r}(f ∘ g ∘ φ^{−1}) . Taking a bundle functor G : Mf → FM of infinite order we produce a bundle functor F (M, N) = M × G(N), F (φ, f) = φ × G(f) of infinite order in the second factor. In [11], we present a complete characterization of all bundle functors F on Mf_{m} × Mf which are of finite order in the first factor.

We have the following new unpublished example.

Example 16. A bundle functor on Mf_{m} × Mf of infinite order in the first factor. For any r = 1, 2, 3, ... we define a vector bundle functor G^{r} : Mf → VB by G^{r}(N) = T (N) × D^{r} and G^{r}(f) = T (f) × id_{D^{r}} : G^{r}(N) → G^{r}(\overline{N}) for any map f : N → \overline{N}, where D^{r} = J_{0}^{r}(R^{m}, R) is the vector space of r-jets at 0 of maps R^{m} → R.

Let d_{r} = dim(G_{0}^{r}(R^{r})). We set

G(N) = ⨁_{k=1}^{+∞} (⋀^{d_{k}} G^{k}(N)) .

For every manifold N, G(N) is a finite dimensional manifold because for k > dim(N) the bundle ⋀^{d_{k}} G^{k}(N) is zero-bundle. Hence the direct sum in the definition of G(N) is in reality a finite sum. For a mapping f : N → \overline{N} the induced mapping G(f) : G(N) → G(\overline{N}) is defined in the natural way from ⋀^{d_{k}} G^{k}(f). In this way we obtain vector bundle functor G on Mf of order 1.

The group G_{m}^{∞} = invJ_{0}^{∞}(R^{m}, R^{m})_{0} of invertible ∞-jets acts on D^{r} by the pull-back. Hence for r = 1, 2, ... we have a group homomorphism H^{r} : G_{m}^{∞} → N E_{vb}(G^{r}) into the group of vector bundle natural equivalences of G by H^{r}(ξ)_{N} : G^{r}(N) → G^{r}(N), H^{r}(ξ)_{N}(v, η) = (v, ξ · η), v ∈ T (N), ξ ∈ G_{m}^{∞}, η ∈ D^{r}, where the dot is the action. Hence for any ξ ∈ G_{m}^{∞} and a manifold N we have a vector natural equivalence H(ξ)_{N} : G(N) → G(N) which is defined in natural way from ⋀^{d_{k}} H^{k}(ξ)_{N} : ⋀^{d_{k}} G^{k}(N) → ⋀^{d_{k}} G^{k}(N). This defines a group homomorphism H : G_{m}^{∞} → N E_{vb}(G).

For any manifold N the corresponding action H of G_{m}^{∞} on G(N) can be factorized by a smooth action of G_{m}^{l_{N}} on G(N) for some finite l_{N}. Hence we can proced similarly as in [11] for above G and H and obtain bundle functor (G, H) on Mf_{m} × Mf. More precisely, we set

(G, H)(M, N) = P^{∞}(M)[G(N), H^{N}] ,

the associated bundle with P^{∞}(M) and fibre G(N). Then (G, H)(M, N) is a finite dimensional manifold diffeomorphic to P^{l_{N}} (M)[G(N)]. We define the source and the target projections α : (G, H)(M, N) → M by α(< j_{0}^{∞}(γ), v >) = γ(0) and β : (G, H)(M, N) → N by β(< j_{0}^{∞}(γ), v >) = y for all j_{0}^{∞}(γ) ∈ P^{∞}(M) and v ∈ G_{y}(N), y ∈ N. Every pair of a local diffeomorphism φ : M → \overline{M} and a map f : N → \overline{N} induce a fibered map (G, H)(φ, f) = (G, H)(M, N) → (G, H)(\overline{M}, \overline{N}) by

(G, H)(φ, f)(< j_{0}^{∞}(γ), v >) =< j_{0}^{∞}(φ ∘ γ), G(f)(v) > .

The bundle functor (G, H) is of order 1 in the second factor. The bundle functor (G, H) is of infinite order in the first factor because the homomorphism H can not be factorized by a group homomorphism G_{m}^{l} → N E(G) with finite l.

Similarly, starting from a bundle functor T̃ : Mf → FM of infinite order instead of T we can produce a bundle functor on Mf_{m} × Mf of infinite order in both factors.

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[17] W. M. Mikulski, Continuity of projections of natural bundles, Ann. Pol. Math. LVII.2 (1992), 105–120.

[18] W. M. Mikulski, There exists a prolongation functor of infinite order, Čas. pest. mat. 114 (1989), 57–59.

[19] W. M. Mikulski, Natural transformations of Weil functors into bundle functors, Suppl. ai Rendiconti del Circolo Matematico di Palermo, Serie II, 1989, 177–191.

[20] W. M. Mikulski, Product preserving bundle functors on fibered manifolds, Arch. Math. Brno 32(4)(1996), 307–316.

[21] W. M. Mikulski Product preserving gauge bundle functors on vector bundles, Colloq. Math. 90(2)(2001), 277–285.

[22] A. Nijenhuis, Natural bundles and their general properties, Diff. Geom. in honor of K. Yano, Tokyo 1972, 317–334.

[23] R. S. Palais and C.-L. Terng, Natural bundles have finite order, Topology 16 (1977), 271–277.

[24] J. Slovák, Bundle functors on fibered manifolds, Annals of Global Anal. and Geom. (1990).

[25] A. Weil, Thèorie des points proches sur les variétés differentiables, Coloque Geom. Diff. Strasbourg 1953, 111–117.

[26] A. Zajtz, The sharp upper bound on the order of natural bundles of given dimension, Bull. Soc. Math. Belgique, 39 (1987), 347–357.

Institute of Mathematics

Jagiellonian University

30–059 Kraków, ul. W. Reymonta 4 POLAND e-mail: mikulski@im.uj.edu.pl

POINCARÉ DUALITY OF TOPOLOGICAL MANIFOLDS

ALEXANDR MISHCHENKO

Abstract

The problem of writing the Hirzebruch formula (see [1]) for topological manifolds for families of representations of the fundamental group collides with two difficulties. The first is related to difinition of rational Pontryagin characteristic classes that demands in particular the Novikov theorem about topological invariance of the rational Pontryagin classes. The second difficulty consists of that the construction of the signature needs a modification of classical construction for topological manifolds. Really, it is impossible to define the Poincare duality as a homomorphism of finite generated differential module since homology by itself is defined using either singular chains or spectrum of open coverings of manifolds. In both cases one should deal with infinite generated modules although homology here turns out to be finite generated spaces.

Here the following construction is possible. Let U = {U_{α}} be a finite covering of the compact manifold X. Let N_{U} be the nerve of the covering U. The nerve N_{U} detemines a finite simplicial polyhedron and hence determines chain and cochain complexes with local system of coefficients defined by a finite dimensional representation ρ of the fundamental group π_{1}(X).

Consider a refining sequence of covering U_{n} = {U_{α}^{n}}, U_{n+1} ≻ U_{n}. This means that U_{α}^{n+1} ⊂ U_{β}^{n} for a proper β = β(α). Hence the function β = β(α) defines a simplicial mapping

π_{n}^{n+1} : N_{U_{n+1}} → N_{U_{n}},

and defines the homomorphism of the homology and cohomology groups

(π_{n}^{n+1})_{∗} : H_{∗} (N_{U_{n+1}}) → H_{∗} (N_{U_{n}}) ,
(π_{n}^{n+1})^{∗} : H^{∗} (N_{U_{n}}) → H_{∗} (N_{U_{n+1}}) .

Here homology and cohomology of the manifold X are defined as limits of homology and cohomology of nerves of the coverings:

H^{∗}(X) = lim_{→} H^{∗} (N_{U_{n}}) ,

H_{∗}(X) = lim_{←} H_{∗} (N_{U_{n}}) .

Then the Poincare duality should defined as a homomorphism D : H^{∗}(X) → H_{∗}(X), generated by the intersection operator with the open cycle of dimension n.

Notice that one can choose the refining sequence of covering in such way that each covering had multiplicity equal to N + 1, N = dim(X). The the Poincare duality can be defined as the intersection operator with the cycle D_{n} ∈ C_{n} \binom{}{Ñ_{U_{n}}}, where (π_{n}^{n+1})_{∗}(D_{n+1}) = D_{n}. There is the following simple argument due to the Mittag-Leffler condition: let C_{∗}^{∞}(X) = lim_{←} C_{∗} (N_{U_{n}}), C_{∞}^{∗}(X) = lim_{→} C^{∗} (N_{U_{n}}). Then H_{∗}(X) = H (C_{∗}^{∞}(X)), H^{∗}(X) = H (C_{∗}^{∞}(X)), where the Poincare homomorphism D is induced by the intersection operator with the cycle D_{∞} = lim_{←}(D_{n}), that is D = H( ∩| D_{∞}).

On the other hand

C_{∗}^{∞}(X) = hom (C_{∞}^{∗}(X); R) ,

that allows to define the signature on the level of a quadratic form defined on the cycle group.

To this end let consider the category C_{0} of countable generated vector spaces over real numbers. Each space V can be considered as a direct limit of countable sequence of finite dimensinal spaces with topology of the direct limit. Then the vector space V^{∗} = hom(V, R) can be represented as a inverse limit of finite dimensional spaces with topology of inverse limit. Denote by hom^{t}(V, R) the space of continuous linear functionals. Then for V ∈ C_{0} one has hom(V, R) = hom^{t}(V, R) where hom^{t}(V^{∗}, R) = V. Let C denote the category with objects being topological space of the form V = V_{1} ⊕ V_{2}^{∗} and morphisms being continuous linear mappings. A non degenerated quadratic form on the object of the category C is defined as a continuous isomorphism

φ : V^{∗} → V, φ^{∗} = φ.

If V = V_{1} ⊕ V_{1}^{∗} and the isomorphism φ defined by a hyperbolic matrix then the quadratic form is called trivial.

þ For non degenerated quadratic form there exists correctly defined invariant signφ which is additive with respect to direct sum, equals to zero on trivial quadratic forms and coincides with classical signature on finite dimensional vector spaces.

The invariant defined above is admissible for constructing of signature of topological manifold with local system of coefficients ([2]).

References

[1] A.S.Mishchenko, The Hirzebruch formula: 45 years of history and the present state of the art, St. Petersburg Math.J., Vol 12, No. 4 (2001), pp 519–533.

[2] M.Gromov, Positive curvature, macroscopic mimension, spectral gaps and higher signatures, Functional Anal. on the Eve of the 21st Century, v. II. Progress in Math., Basel–Boston: Birkhauser,Vol. 132 (1995).

Moscow State University

Department of Mathematics

119 992 GSP–2 Moscow

RUSSIA

   http://higeom.math.msu.su/asmish

e-mail: asmish@higeom.math.msu.su

MINIMAL NILPOTENT ALGEBRAS IN GOURSAT FLAGS OF LENGTHS NOT EXCEEDING 6

PIOTR MORMUL

1 Nilpotent approximation of a distribution at a point.

For any distribution D of rank d on an n–dimensional, smooth or real analytic, manifold M (i. e., rank–d subbundle in the tangent bundle T M) its small flag is the nested sequence

V_{1} ⊂ V_{2} ⊂ V_{3} ⊂ ⋯

of modules (or: presheaves of modules) of vector fields (of the same category as M) tangent to M: V_{1} = D, V_{j+1} = V_{j} + [D, V_{j}] for j = 1, 2, … The small growth vector at p ∈ M is the sequence (n_{j}) of linear dimensions at p of the modules V_{j}: n_{j} = dimV_{j}(p). Naturally, n_{1} = d independently of p.

D is completely nonholonomic when at every point of M its small growth vector attains (sooner or later) the highest value n = dimM. Once this value attained, we truncate the vector after the first appearance of n in it. The length d_{NH} of thus truncated vector is called the nonholonomy degree of D at p.

In the theory that we recall (cf. [1], [AGS], [BS], [B]; this list of references is not complete) important are the weights w_{i} related to the small flag at a point: w_{1} = ⋯ = w_{d} = 1, w_{d+1} = ⋯ = w_{n_{2}} = 2 (no value 2 among them when n_{2} = d), and generally w_{n_{j}+1} = ⋯ = w_{n_{j+1}} = j + 1 (no value j + 1 among them when n_{j} = n_{j+1}) for j = 1, 2, …

Definition 1. For a completely nonholonomic distribution D on M, coordinates z_{1}, z_{2}, …, z_{n} around p ∈ M are linearly adapted at p when D(p) = (∂_{1}, …, ∂_{d}), V_{2}(p) = (∂_{1}, …, ∂_{d}, …, ∂_{n_{2}}), and so on until V_{d_{NH}}(p) = (∂_{1}, …, ∂_{n}) = T_{p}M. Here and in the sequel we skip writing ‘span’ before a set of v. f. generators. For such linearly adapted coordinates we define their weights w(z_{i}) = w_{i}, i = 1, …, n.

On the other hand, having a completely nonholonomic D, every smooth function f on M near p has its nonholonomic order wrt D at p (+∞ is not excluded). It is the minimal number of differentiations of f along the local generators of D that give at p a nonzero result. It follows directly from the above definitions that, for linearly adapted coordinates, their nonholonomic orders do not exceed their weights. Linearly adapted coordinates z_{1}, …, z_{n} are adapted (or: privileged ) when the nonholonomic order of z_{i} equals w(z_{i}), i = 1, …, n. (In particular, adapted coordinates must vanish at p; one says that they are centered at p.)

It is not so quick, but true, that adapted coordinates always exist, and can even be algorithmically constructed from any à priori given coordinates, even in a polynomial way, as is done for inst. in [B]. They are not unique, there remains plenty of liberty behind the requirement that the nonholonomic orders (of linearly adapted coordinates) be maximal possible. ln adapted coordinates it is purposeful to attach quasihomogeneous weights also to monomial vector fields (this definition goes back to the 1970’s, to the theory of differential operators),

w(z_{i_{1}} ⋯ z_{i_{k}}∂_{j}) = w(z_{i_{1}}) + ⋯ + w(z_{i_{k}}) − w(z_{j}) .  (1)

Proposition 3. Every smooth vector field X with values in D has in its Taylor expansion in arbitrary coordinates adapted for D only terms of weights not smaller than −1 that can be grouped in homogeneous summands X = X^{(−1)} + X^{(0)} + X^{(1)} + ⋯

(superscripts mean the weights defined by (1)). We denote by X̂ the lowest (‘nilpotent’) summand X^{(−1)}. That is, X̂ = X^{(−1)}. When a distribution D has around p local generators (vector fields) X_{1}, …, X_{d}, then

Definition 2. The distribution D̂ = (X̂_{1}, …, X̂_{d}) is called the nilpotent approximation of D at p.

This object D̂ is invariantly defined, independently of the used adapted coordinates, see Prop. 5.20 in [B]. Its basic properties are

Proposition 4. The nilpotent approximation D̂ of D has at p the same small growth vector as D (and hence the same nonholonomy degree d_{NH}). Moreover, the real Lie algebra generated by X̂_{1}, …, X̂_{d} is a nilpotent Lie algebra of nilpotency order d_{NH}

(the nilpotency order of a Lie algebra is the minimal number of multiplications in that algebra yielding always zero).

Definition 3 ([AG]). Distribution D is strongly nilpotent at a point p when its germ at p is equivalent to its nilpotent approximation D̂ at p. ([HLS], [M4]) Distribution D is weakly nilpotent at a point p when D possesses, locally around p, a basis generating over ℝ a nilpotent (real) Lie algebra of vector fields.

In the founding work [HLS], and posterior control theory literature, weakly nilpotent distributions are called nilpotentizable. Nilpotent approximations of distributions, and in particular strongly nilpotent distributions play a growing role in sub-riemannian geometry ([B], [AG], [5]).

2 Kumpera-Ruiz algebras of Goursat distributions.

In the sequel we deal with Goursat distributions – a rather restricted class of objects for which preliminary (local) polynomial normal forms of [KR] exist with real parameters only, and no functional moduli. Their definition requires that the sequence of consecutive Lie squares of the original rank–2 subbundle of T M consist of regular distributions of ranks 3, 4,… until n = dimM. In the abstract [M3] we recalled a basic partition of Goursat germs into disjoint geometric classes encoded by words of length n − 2 over the alphabet G, S, T, with two first letters always G and such that never a T goes directly after a G. Their construction was done by Montgomery and Zhitomirskii; it is reproduced in Sec. 1.3 of [M2]. (Implicitly these classes are already present in a pioneering work [J], in which the author uses a trigonometric presentation of Goursat objects. Another way of constructing them has been proposed in [PR1] where they are called singularity types.)

In dimension 4 there is but one class GG, in dimension 5 – only GGG and GGS, in dimension 6 – GGGG, GGSG, GGST, GGSS, GGGS.

The union of all geometric classes of fixed length with letters S in fixed positions in the codes is called, after [MZ], a Kumpera-Ruiz class of Goursat germs of that corank. For inst., in dimension 6 the two geometric classes GGSG and GGST build one KR class ∗ ∗ S∗.

Passing to nonholonomy degrees, Jean was able to compute them for objects showing up in his trigonometric presentation. That, in view of Thm. 4.1 of [BH] (see also [MZ], [PR1]) and after putting geometric classes in relief in Jean’s approach, does for all G. germs.

Proposition 5 ([J]). In dimension n, the nonholonomy degree d_{NH} of any Goursat germ in the geometric class C, equals the last term b_{n−2} in the sequence b_{1}, b_{2}, …, b_{n−2} defined only in terms of C: b_{1} = 2, b_{2} = 3, b_{j+2} = b_{j} + b_{j+1} when the (j + 2)-th letter in C is S, b_{j+2} = 2b_{j+1} − b_{j} when the (j + 2)-th letter in C is T, b_{j+2} = 1 + b_{j+1} when the (j + 2)-th letter in C is G.

The mentioned polynomial (local) presentations of G. objects were not used in [J], but what are they? The essence of the contribution [KR], given in the notation of vector fields and taking into account posterior works, is as follows. We construct a (not unique) rank–2 distribution on (ℝ^{n}(x^{1}, …, x^{n}), 0) departing from the code of a geometric class C.

When the code starts with precisely s letters G, one puts Y^{1} = ∂_{1}, Y^{2} = Y^{1} + x^{3}∂_{2} , …,

Y^{s+1} = Y^{s} + x^{s+2}∂_{s+1}. When s < n − 2, then the (s + 1)th letter in C is S. More generally,

if the mth letter in C is S, and Y^{m} is already defined, then

Y^{m+1} = x^{m+2}Y^{m} + ∂_{m+1} .

But there can also be T’s or G’s after an S. If the mth letter in C is not S, and Y^{m} is already defined, then

Y^{m+1} = Y^{m} + (c^{m+2} + x^{m+2})∂_{m+1} ,

where a real constant c^{m+2} is not absolutely free but

• equal to 0 when the mth letter in C is T,

• not equal to 0 when the mth letter is G going directly after a string ST… T (or after a short string S).

Now, on putting X = ∂_{n} and Y = Y^{n−1}, and understanding (X, Y) as the germ at 0 ∈ ℝ^{n},

Theorem 3 ([KR]). Any Goursat germ D on a manifold of dimension n, sitting in a geometric class C, can be put (in certain local coordinates) in a form D = (X, Y), with certain constants in the field Y corresponding to G’s past the first S in C.

Definition 4. The real Lie algebra generated by X and Y is called the KR algebra of the germ D. This algebra does not depend on the choice of coordinates in Thm. 3, although so do the constants in Y. Its nilpotency order is denoted by O_{KR}. (A short analysis shows that this algebra depends solely on the Kumpera-Ruiz class of D.)

In 2000 we proved (cf. Rem. 1 in [M1])

Theorem 4 ([M4]). The KR algebra of any Goursat distribution in dimension n, locally around a point belonging to a geometric class C, is nilpotent of nilpotency order d_{n−2}, where d_{n−2} is the last term in the sequence d_{1}, d_{2}, …, d_{n−2} defined only in function of the Kumpera-Ruiz class subsuming C: d_{1} = 2, d_{2} = 3, d_{j+2} = d_{j} + d_{j+1} when the (j + 2)-th letter in C is S, and otherwise d_{j+2} = 2d_{j+1} − d_{j}.

Note that in [PR2] a result stating just the nilpotency of the same algebras (the existence of finite nilpotency orders for them) is given.

Definition 5. We call tangential geometric classes whose codes possess letters G only in the beginning, before the first S (if any) in the code. Tangential Goursat germs are those sitting in the tangential classes.

Example 1. Up to dimension 5 all geometric classes are tangential. The first, and unique non-tangential class in dimension 6 is GGSG. In dimension 7 there are eight tangential classes and five non-tangential: GGSGG, GGSTG, GGSGS, GGGSG, GGSSG. In dimension 8 there are sixteen tangential and eighteen non-tangential geometric classes (cf. Thm. 5 below).

Tangential classes become clearly visible when one uses the polynomial presentation.

Observation 1. A Goursat germ D is tangential ⟺ in any KR presentation (X, Y) of D there is no non-zero constant

(so tangential germs are easily given local models with no parameters).

3 Minimality of KR algebras in several geometric classes.

The comparison of the formulas for the nonholonomy degree d_{NH} (Prop. 5) with the ones for nilpotency order O_{KR} (Thm. 4) yields that in each fixed geometric class C a). d_{NH} ≤ O_{KR}, and b). d_{NH} = O_{KR} only when C is tangential. Thus within tangential classes the KR algebras cannot be improved in the sense of lowering nilpotency orders. (Also – as is visible in the proof of Thm. 4 in [M4] – the germs in tangential classes are all strongly nilpotent, and Kumpera-Ruiz coordinates of Thm. 3 are already adapted.) How is it in non-tangential classes? Do there exist there Goursat germs with better nilpotent bases – with lower nilpotency orders?

This question makes sense from dimension 6 onwards (see Ex. 1). We announce below the full answer in dimensions 6 and 7 (addressing all non-tangential classes in these dimensions), and a partial one in dimension 8 (addressing eight out of eighteen non-tangential classes).

Theorem 5. A. In dimension 6, for germs in the class GGSG the nilpotency order O_{KR} = 7 is minimal among all possible local nilpotent bases, despite the fact that d_{NH} = 6 for these germs.

B. In dimension 7, for germs in the classes:

geometric class d_{NH} O_{KR}
GGSGG  7  9
GGSTG  8  9
GGSSG  9  11
GGSGS  11  12
GGGSG  8  10

their respective KR algebras are of minimal possible nilpotency order.

C. In dimension 8, for germs in the non-tangential classes:

geometric class d_{NH} O_{KR}
GGSGGG  8  11
GGSTGG  9  11
GGSTTG  10  11
GGSSGS  17  19
GGSGSG  12  17
GGSTSG  13  17
GGSGST  16  17
GGGSGS  15  17

their respective KR algebras are of minimal possible nilpotency order, too.

We do not yet know the answer in the remaining ten non-tangential classes in dimension 8.

Corollary 1. In dimension 6 and 7, for all Goursat germs in the non-tangential geometric classes, their nilpotent approximations are not equivalent to the departure germs. The same concerns the germs in dimension 8 sitting in the eight non-tangential classes listed in Thm. 5, C. It is so because of the second property of nilpotent approximations recalled in Prop. 4. Thus those germs are not strongly nilpotent.

The weak form of nilpotency – possession of a nilpotent basis – appears thus much weaker than the strong form of nilpotency of a distribution germ. In [M4] it is conjectured that the pattern emerging from Thm. 5 is valid for all non-tangential geometric classes in all dimensions. Implying that, conjecturally, the notions ‘tangential’ and ‘strongly nilpotent’ simply coincide.

References

[AGS] A. A. Agrachev, R. V. Gamkrelidze, A. V. Sarychev; Local invariants of smooth control systems. Acta Appl. Math. 14 (1989), 191 – 237.

[AG] A. A. Agrachev, J–P Gauthier; On subanalyticity of Carnot–Caratheodory distances, preprint 25/2000/M, SISSA, Trieste.

[B] A. Bellaïche; The tangent space in sub-Riemannian geometry. In: A. Bellaïche, J-J. Risler (eds.), Sub-Riemannian Geometry, Birkhäuser, Basel – Berlin 1996, 1 – 78.

[BS] R. M. Bianchini, G. Stefani; Graded approximations and controllability along a trajectory. SIAM J. Control Optimiz. 28 (1990), 903 – 924.

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

[H] H. Hermes; Nilpotent approximation of control systems and distributions. SIAM J. Control Optimiz. 24 (1986), 731 – 736.

[HLS] H. Hermes, A. Lundell, D. Sullivan; Nilpotent bases for distributions and control systems. J. Diff. Eqns 55 (1984), 385 – 400.

[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.

[KR] 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.

[MZ] R. Montgomery, M. Zhitomirskii; Geometric approach to Goursat flags. Ann. Inst. H. Poincaré – AN 18 (2001), 459 – 493.

[M1] P. Mormul, Goursat flags: classification of codimension–one singularities. J. Dyn. Control Syst. 6 (2000), 311 – 330.

[M2] 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

[M3] P. Mormul, Geometry of Goursat flags and their singularities of codimension 2. Abstract submitted to the 3rd Conference GEOMETRY and TOPOLOGY of MANIFOLDS, Krynica 2001.

[M4] P. Mormul, Goursat distributions not strongly nilpotent in dimensions not exceeding seven. To appear in Proceedings NCN4, Sheffield 2001.

[PR1] 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.

[PR2] W. Pasillas–Lépine, W. Respondek; Nilpotentization of the kinematics of the n-trailer system at singular points and motion planning through the singular locus. Int. J. Control 74 (2001), 628 – 637.

[S] Y. Sachkov; On nilpotent sub-Riemannian (2, 3, 5) problem. Workshop Math. Control Th. and Robotics, Trieste 2000.

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

INVARIANT KÄHLER POLARIZATIONS AND GEOMETRY OF VECTOR FIELDS

IHOR MYKYTYUK

Abstract

Let G/K be a Riemannian symmetric space. Let 𝔤 = 𝔨⊕𝔪 be the corresponding invariant splitting of the Lie algebra 𝔤 of G. Geometric constructions which come from geometric quantization naturally lead to complex structures defined on the punctured cotangent bundle T_{0}^{∗}(G/K) = T^{∗}(G/K) − {zero section}. Such structure J_{S} for the spheres was found by Souriau. Later it was observed by Rawnsley, that the length function is strictly plurisubharmonic with respect to the above complex structure J_{S} and defines the Kähler metric on T_{0}^{∗}S^{n} with the canonical symplectic form Ω as the Kähler form. He also observed that J_{S} is invariant with respect to the Hamiltonian flow of the length function (the normalized geodesic flow) and used the Kähler polarization J_{S} to quantize the geodesic flow on the spheres.

Our purpose is to describe G-invariant Kähler structures on T^{∗}(G/K). Denote by ⟨ , ⟩ the scalar product on 𝔪 defining the Riemannian metric on G/K. Let S^{+}(𝔪^{ℂ}) be the set of all invertible symmetric (with respect to ⟨ , ⟩) operators s : 𝔪^{ℂ} → 𝔪^{ℂ} such that Re s > 0 (positive definite).

Theorem.There is one-to-one correspondence between the set of all G-invariant Kähler structures {(J, Ω)} on T^{∗}(G/K) and the set {P } of all smooth mappings P : 𝔪 → S^{+}(𝔪^{ℂ}), w ↦ P_{w} such that 1) (Ad_{k} P Ad_{k^{−1}})|𝔪 = P_{Ad_{k} w}, ∀w ∈ 𝔪, k ∈ K; 2) [P_{w}(ξ), P_{w}(η)](w) = −[w, [ξ, η]] on 𝔪, ∀ξ, η ∈ 𝔪, where we consider the maps w ↦ P_{w}(ξ), w ↦ P_{w}(η) as vector fields on 𝔪.

It is described all mappings P satisfying conditions of Theorem in the following cases:

(1) if rank G/K = 1, dim G/K ≥ 3; (2) if the corresponding Kähler structure (J, Ω) is invariant under the normalized geodesic flow. For Hermitian symmetric spaces G/K of rank ≥ 2 a family of Kähler structures (of mappings P) is constructed. As an application of these results we obtain 1) a description of the curvature tensor of the symmetric space F_{4}/Spin(9) in terms of its Spin(9)-structure; 2) new results concerning the hyper-Kähler metric on T^{∗}(G/K) if G/K is a Hermitian symmetric space of rank ≥ 1 (in this case the natural holomorphic-symplectic structure underlies the hyper-Kähler metric, whose restriction to G/K is the given homogeneous metric).

Applied Mathematics Department

State University ”Lvi’v Polytechnica”

S. Bandera St. 12

79013, Lviv

UKRAINE

e-mail: viva@iapmm.lviv.ua

LOCAL STRUCTURES OF KV-ALGEBROIDS AND LIE ALGEBROIDS

MICHEL NGUIFFO BOYOM and ROBERT WOLAK

Abstract

The word manifold is used for smooth or complex analytic manifold. We deal with connected and paracompact manifolds. For a given manifold M F (M, K) stands for the associative K-algebra of K-valued smooth functions if K = ℝ (resp. the sheaf of complex analytic functions if K = ℂ).Geometric objects considered on M are either smooth or complex analytic according to K := ℝ or ℂ . Given a vector bundle A on M sect(A) stands for the F (M, K)-module of sections of A. A Koszul-Vinberg algebroid (resp Lie algebroid ) on M is couple (A, a) where A is a vector bundle on M and a is a vector bundle M-homomorphism from A to T M (T M is the holomorphic tangent bundle when K = ℂ) such that:

i) sect(A) is endoved with a structure of Koszul-Vinberg algebra (sect(A), . ) (resp. structure of Lie algebra (sect(A), [ , ]);

ii) given S, S′ in sect(A) and f in F (M, K) one has (fS).S′ = f(S.S′) and S.(fS′) = f(S.S′) + (a(S)f)S′, (resp. [fS, S′] = f[S, S′] + (a(S)f)S′).

The purpose of the confernce is to expose a decomposition theorem ( normal forms ) of KV -algebroids ( resp. Lie algebroids ) and a local classification theorem as well.

Michel NGUIFFO BOYOM Departement de Mathématiques Université Montpellier 2 FRANCE e-mail: boyom@math.univ-montp2.fr

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

EXTENDED POINCARÉ PARASUPERGROUP WITH CENTRAL CHARGES

ANATOLY NIKITIN

Abstract

Irreducible represemtations of thr extended Poincaré parasuperalgebra with non-trivial central charges are described. Linear and non-linear models invariant w.r.t. the related parasupergroup are discussed.

Institute of Mathematics of Nat. Acad. Sci. of Ukraine 3 Tereshchenkivska St, 01601 Kiev–4 UKRAINE e-mail: nikitin@imath.kiev.ua

http://www.imath.kiev.ua/~nikitin/

THE HOLOMORPHIC PSEUDOSYMMETRY OF KÄHLER MANIFOLDS

ZBIGNIEW OLSZAK

Abstract

The holomorphic pseudosymmetry is a natural generalization of the semisymmetry, in particular, the local symmetry.

A Kähler manifold M(J, g) is said to be holomorphically pseudosymmetric if its Riemann curvature R satisfies the condition

R · R = f R̃ · R,

where f is a certain function on M and R̃ is the derivation of the tensor algebra generated by the curvature type endomorphism

R̃(X, Y) = X ∧ Y + (JX) ∧ (JY) − 2g(JX, Y)J.

Basing on the Lichnerowicz’s integral formulas, we can investigate the holomorphic pseudosymmetry of compact Kähler manifolds. The main result is the following:

Let M be a compact Kähler manifold with constant scalar curvature. If M is holomorphically pseudosymmetric with f ≥ 0, then it is locally symmetric.

Institute of Mathematics Wrocław University of Technology Wybrzeże Wyspiańskiego 27, 50–370 Wrocław POLAND e-mail: olszak@im.pwr.wroc.pl

GEOMETRIC ASPECTS OF INVARIANTS OF FINITE TYPE OF KNOTS IN S^{3}

LEONID PLACHTA

Abstract

The knot invariants of finite type (or Vassiliev invariants) give a coherent and systematic way to study the polynomial invariants of knots in S^{3}. However the geometric nature and geometric understanding of both the polynomial and the Vassiliev invariants remain missing. That is, it is unclear what kind of geometric knot information is carried by these invariants. In the present talk, we briefly review the well-known facts on this subject and report some new results.

The two knots are called n-equivalent, where n ≥ 1, if they cannot be distinguished by the Vasssiliev invariants of order ≤ n, the invariants taking values in any abelian group. The ralation of n-equivalence on knots in S^{3} has been characterized by Habiro [1], in terms of C_{n+1}-moves on knots, and Stanford [5], in terms of the pure braid (n + 1)-commutators. A knot is called n-trivial, if it is n-equivalent to the trivial one. In [2], Kalfagianni and Lin initiated the study of knot invariants of finite type in S^{3} via geometric methods. They interpret these invariants as obstructions to a knot’s bounding a regular Seifert surface with certain properties. More precisely, in order to describe the classes of n-equivalent knots, they introduced for each n > 1 the classes of n-hyperbolic, n-elliptic and n-parabolic knots, and showed that all they are n-trivial. It is unknown however whether the class consisting of all n-hyperbolic, n-elliptic and n-parabolic knots coincides with the one of n-trivial knots. They asked if furthermore, (n − 1)- and (n − 2)-hyperbolic knots are n-trivial. We show that in the first case the answer to the question is affirmative, while in the second case this is negative. More precisely, we show that all (n − 1)-hyperbolic, n > 2, are n-trivial and that for each odd integer n > 3 there is an (n − 2)-hyperbolic knot which is not n-trivial. Note that any n-hyperbolic and n-elliptic knots have the trivial Alexander polynomial. We introduce the notion of n-band equivalent knots and study the relationship between the class of all n-hyperbolic and n-elliptic knots and the class of knots, n-band equivalent to the trivial one. We discuss also the problem of how can change the group of a knot, after an application of a simple C_{n}-move to it, or equivalently, after an insertion of a pure braid n-commutator p_{σ} [3] in the knot. We provide a sufficient condition under which a simple C_{n}-move on a knot does not affect the Alexander polynomial of it [4]. Several problems concerning the geometric properties of invariants of finite order are also raised.

References

[1] K.Habiro, Claspers and finite type invariants of links, Geom. and Top. 4, 2000, 1-83.

[2] E.Kalfagianni and X.-S.Lin, Regular Seifert surfaces and Vassiliev knot invariants, Preprint 1998, math. GT/9804032S.

[3] L.Plachta, C_{n}-moves, braid commutators and Vassiliev knot invariants (submit. to Compositio Math.)

[4] L.Plachta, n-trivial knots and the Alexander polynomial (in preparation)

[5] T.Stanford, Vassiliev invariants and knots modulo pure braid subgroups, Preprint 1998, math.GT/9805092.

Institute of Applied Problems of Mechanics and Mathematics of NAS of Ukraine UKRAINE e-mail: igor@iapmm.lviv.ua

ON HIGER GEOMETRY AND INDUCED OBJECTS ON SUBSPACES

MARCELA POPECSU and PAUL POPESCU

Abstract

Besides a theory of higher order Finsler and Lagrange spaces, a dual theory of higher order Hamilton spaces was only recentely studied using the bundles of accelerations. In this paper we investigate the possibility to use these ideas in a more general setting. A recursive definition of higher order bundles defined by an affine bundle E and a vector pseudo-field on E can be considered, obtaining the acceleration bundles as a particular case. The case of non-holonomic spaces is effectively studied. A dual theory between lagrangians and hamiltonians (via Legendre transformations) is considered using affine bundles. A canonical way to induce a hamiltonian on an affine subbundle is also given.

Department of Algebra and Geometry

University of Craiova

13, ’Al.I.Cuza’ st., Craiova, 1100

ROMANIA

e-mail: marcelapopescu@email.com, paulpopescu@email.com

A SMOOTH STRUCTURE ON LOCALLY COMPACT TOPOLOGICAL GROUP

TOMASZ RYBICKI

Abstract

It is shown that any connected locally compact topological group (or, more generally, any P L-group) carries a smooth structure. Here a smooth structure is determined by a set of smooth curves with some conditions specified. It is observed that the Lie algebra of a locally compact group is identified with the Lie algebra assigned to a smooth group structure. Further properties are exhibited.

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

PROPERTIES OF THE CURVATURE OF KÄHLER–NORDEN MANIFOLDS

KARINA SŁUKA

Abstract

Let M be a smooth 2n-dimensional manifold with an almost complex structure J and a pseudo-Riemannian anti-Hermitian metric g, i.e.

J^{2} = −I, g(JX, JY) = −g(X, Y).

If additionaly ∇J = 0, where ∇ is the Levi-Civita connection of g, then the triple (M, J, g) form a Kähler-Norden manifold. A structure of this kind is also called an almost complex structure with B-metric, or an anti-Kählerian structure.

We have considered the classes of Kähler-Norden manifolds with recurrent curvature, recurrent holomorphic projective curvature, or recurrent conformal curvature.

We are also interested in those Kähler-Norden manifolds satisfying the conditions of semisymmetry or more generally the conditions of pseudosymmetry type.

Institute of Mathematics Wrocław University of Technology Wybrzeże Wyspiańskiego 27, 50–370 Wrocław POLAND e-mail: sluka@mazur.im.pwr.wroc.pl