TANGENTIAL CHERN-WEIL HOMOMORPHISM ^{*}

Jan KUBARSKI

Institute of Mathematics, Technical University of Łódź,

PL-90-924 Łódź, Al.Politechniki 11, Poland

Abstract

Moore and Schochet gave the Chern-Weil homomorphism of a vector bundle 𝔣 over a foliated space (M, F), measuring the nonexistence of partially flat covariant derivatives. We look at this problem (restricting our interest to vector bundles over foliated manifolds) from the point of view of nontransitive Lie algebroids. We use – for our problem – the Chern-Weil homomorphism of the regular Lie algebroid coming from the Lie algebroid A(𝔣) of 𝔣 by restricting it to the elements whose anchors are tangent to the foliation F. Our observations lead to the conjecture that we can sometimes obtain some essentially new kinds of characteristic classes (with respect to the construction of Moore-Schochet) called singular.

1 Introduction

1.1 Category of regular Lie algebroids

Let F be a C^{∞} constant dimensional and involutive distribution on a C^{∞} Hausdorff paracompact connected manifold M. By a regular Lie algebroid over a foliated manifold (M, F) [5] (see also [3], [4], [7]) we mean a system

(A, [[·, ·]], γ)

consisting of a vector bundle A over M and mappings

[[·, ·]] : SecA × SecA → SecA, γ : A → T M,

such that:

*This paper is in final form and no version of it will be submitted for publications elsewhere

1) (SecA, [[·, ·]]) is a real Lie algebra;

2) γ, called by K.Mackenzie [7] an anchor, is a homomorphism of vector bundles, and Imγ = F;

3) Secγ :SecA → X (M), ξ ↦ γ ∘ ξ, is a homomorphism of Lie algebras;

4) [[ξ, f · η]] = f · [[ξ, η]] + (γ ∘ ξ)(f) · η, ξ, η ∈SecA, f ∈ C^{∞}(M).

A Lie algebroid (A, [[·, ·]], γ) is called transitive if F = T M.

Let (A, [[·, ·]], γ) and (A′, [[·, ·]]′, γ′) be two Lie algebroids on the same manifold M. By a homomorphism

H : (A, [[·, ·]], γ) → (A′, [[·, ·]]′, γ′)

between them we mean a homomorphism H : A → A′ of vector bundles over id_{M} : M → M such that:

1) γ′ ∘ H = γ;

2) Sec H : Sec A′ → Sec A, ξ ↦ H ∘ ξ, is a homomorphism of Lie algebras.

A homomorphism H of Lie algebroids induces a homomorphism of the associated exact Atiyah sequences

0 ⟶ ggg ↪ A ⟶^{γ} F ⟶ 0
↓ H^{+} ↓ H  ↓
0 ⟶ ggg′ ↪ A′ ⟶^{γ′} F′ ⟶ 0 .

A trivial Lie algebroid is defined to be one isomorphic to (T M ×𝔤, [[·, ·]], 𝔭𝔯_{1}) where 𝔤 is a finite-dimensional Lie algebra and the bracket is given by the formula

[[(X, σ), (Y, η)]] = ([X, Y], L_{X}η − L_{Y} σ − [σ, η]),

X, Y ∈ X (M), σ, η ∈ C^{∞}(M; 𝔤). Every transitive Lie algebroid over a manifold diffeomorphic to ℝ^{⋉} is trivial [6], [7].

1.2 Sources of Lie algebroids

Differential Geometry has discovered many objects which determine Lie algebroids, such as, for example, differential groupoids, principal bundles, vector bundles, transversally complete foliations, nonclosed Lie subgroups, Poisson manifolds, some complete closed pseudogroups, and actions of Lie groups on manifolds.

On the other hand, if (A, [[·, ·]], γ) is a transitive Lie algebroid on M and (M, F) is a foliated manifold, then A^{F} := γ^{−1}[F] ⊂ A forms, in an evident manner, a regular Lie algebroid over (M, F). For example, a principal bundle or a vector bundle over a foliated manifold determines an object which is the fundamental tool serving our problem. For the other examples, see ”Characteristic homomorphisms of regular Lie algebroids”, Poster, European Congress of Mathematics, Paris, 1992, contained in [6].

1.3 Representations and invariant cross-sections

Let 𝔣 be a vector bundle over M. We recall [5] (see also [7]) that the fibre A(𝔣)_{|x} of the Lie algebroid A(𝔣) of 𝔣 over x consists of all linear homomorphisms l :Sec𝔣 → 𝔣_{|x}, called 𝔣-vectors, for which there exists a vector u ∈ T_{x}M such that

l(f · ν) = f(x) · l(ν) + u(f) · ν(x), f ∈ C^{∞}(M), ν ∈ Sec𝔣.

u is the anchor of l and is denoted by q(l). A cross-section ξ ∈ Sec A(𝔣) defines a covariant differential operator L_{ξ} : Sec 𝔣 → Sec 𝔣, L_{ξ}(ν)(x) = ξ_{x}(ν), x ∈ M; L_{[[ξ,η]]} = [L_{ξ}, L_{η}] = L_{ξ} ∘ L_{η} − L_{η} ∘ L_{ξ} [7].

The Lie algebroid A(𝔣) is (locally) described in the following:

Theorem 1.3.1 [5; 5.4.4] Let ψ : U × V → p^{−1}[U] be a local trivialization of a vector bundle 𝔣 (with V as a typical fibre). For ν ∈ Sec 𝔣, denote by ν_{ψ} the function U ∋ x ↦ ψ_{|x}^{−1}(ν_{x}) ∈ V. Then the mapping

ψ̄ : T U × End(V) ⟶ A(𝔣)_{|U}

such that

ψ̄(v, a)(ν) = ψ_{|x}(∂_{v}(ν_{ψ}) + a(ν_{ψ}(x)))

when v ∈ T_{x}U and a ∈ End(V), is an isomorphism of Lie algebroids. □

By a representation of a Lie algebroid A on 𝔣 [5], [7] we mean a homomorphism T : A → A(𝔣) of Lie algebroids. A cross-section ν ∈ Sec 𝔣 is called T –invariant if

T (v)(ν) = 0 for all v ∈ A.

Denote by (Sec 𝔣)_{I^{∘}(T)} (or (Sec 𝔣)_{I^{∘}} when there is no misunderstanding) the space of all T -invariant cross-sections of 𝔣.

1.4 Connections, curvature and the Chern-Weil homomorphism of regular Lie algebroids

Let A be any regular Lie algebroid over a foliated manifold (M, F). Any splitting λ : F → A of the exact Atiyah sequence

0 ⟶ ggg ↪ A ⟶^{γ} F ⟶ 0 ; ⟵^{λ}

is called a connection in A. The tangential differential form

Ω ∈ Ω_{F}^{2} (M,ggg) = Sec(⋀^{2} F^{⋆} ⨂ggg)

defined by

Ω(X, Y) = λ∘[X, Y] − [[λ∘X, λ∘Y]], X, Y ∈ Sec F,

is called the curvature tensor of λ. If A = A(P)^{F}, P being a principal bundle and A(P) = T P/G – its Lie algebroid [4], [7], then there is a bijection between connections in A and partial connections in P over F.

By the adjoint representation ad_{A} : A → A(ggg) of a regular Lie algebroid A we mean the one defined by ad_{A}(v)(σ) = [[ξ, σ]](x) where ξ is an arbitrarily taken cross-section of A such that ξ(x) = v ∈ A_{|x} and σ ∈ Secggg. The mapping ad_{A} induces canonically a representation ⋁^{k} ad_{A}^{♮} of A on the symmetric power ⋁^{k} ggg^{⋆} denoted, for short, also by ad_{A}. Let I^{∘k}(A) := (Sec(⋁^{k} ggg^{⋆}))_{I^{∘}} be the space of invariant elements with respect to this representation. I^{∘}(A) := ⨁^{k≥0} I^{∘k}(A) forms an algebra. We recall that Γ ∈ I^{∘k}(A) if and only Γ ∈ Sec(⋁^{k} ggg^{∗}) and

∀_{ξ∈Sec A} ∀_{σ_{1},... ,σ_{k}∈Secggg} ( (γ∘ξ)⟨Γ, σ_{1} ∨ … ∨ σ_{k}⟩
= ∑_{i=1}^{k}⟨Γ, σ_{1} ∨ … ∨ [[ξ, σ_{i}]] ∨ … ∨ σ_{k}⟩ )

Theorem 1.4.1 [5] For Γ ∈ I^{∘k}(A) and the curvature tensor Ω of a connection in A, the real tangential form

β(Γ) = \frac{1}{k!}⟨Γ, \underbrace{Ω ∨ ⋯ ∨ Ω}_{k times}⟩ ∈ Ω_{F}^{2k}(M)

is closed. The mapping

h_{A} : I^{∘}(A) → H_{F} (M), Γ ↦ [β(Γ)],  (1.4.1)

called the Chern-Weil homomorphism of A, is a homomorphism of algebras independent of the choice of a connection. □

Remark 1.4.2 H_{F} (M) in the above theorem denotes the tangential cohomology space of (M, F) being the cohomology space of the complex (Ω_{F} (M), d^{F}) of real tangential differential forms, where d^{F} is the standard differentiation defined in an elementary way in terms of local coordinates [8] or, equivalently, by the global formula, the same as for usual real differential forms.

Remark 1.4.3 Homomorphism (1.4.1) for the Lie algebroid A(P) of a connected principal bundle P = P (M, G) agrees with the classical Chern-Weil homomorphism h_{P} : I(G) → H_{dR}(M) of P [2] in the sense that there exists a natural isomorphism α : I(G) → I^{∘}(A) of algebras such that

h_{A(P)} ∘ α = h_{P}

We pay our attention to the fact that this holds although in the Lie algebroid A(P) there is no direct information about the structure Lie group G of P which may be disconnected !

Remark 1.4.4 There are transitive Lie algebroids which do not come from principal bundles but have nontrivial Chern-Weil homomorphisms (Lie algebroids of some transversally complete foliations have this property [5] ).

1.5 The inverse image of a regular Lie algebroid and of a representation

In [5] there is a construction of some important (from the technical point of view) notion of the inverse image f^{∧}A of a regular Lie algebroid A over (M, F) by a morphism f : (M′, F′) → (M, F) of the category of foliated manifolds (i.e. a smooth mapping f : M′ → M such that f_{⋆}[F′] ⊂ F). In this paper we shall only take the above notion for the inclusion i_{L} : (L, T L) ↪ (M, F) where L is a leaf of the foliation F. Then i_{L}^{∧}(A) is the transitive Lie algebroid for which

1) the total space is equal to T L ×_{i_{L⋆},γ} A = {(v, w) ∈ T L × A; v = γw};

2) the projection pr_{1} : T L ×_{i_{L⋆},γ} A → T L is the anchor;

3) the Lie bracket in Sec(i_{L}^{∧}A) is defined uniquely by demanding that, for ξ, η ∈ Sec A,

[[(γ ∘ ξ | L, ξ | L), (γ ∘ η | L, η | L)]] = ([γ ∘ ξ, γ ∘ η] | L, [[ξ, η]] | L).

Clearly, the total space of the Lie algebroid i_{L}^{∧}(A) is isomorphic to the restriction A_{|L} via the linear isomorphism A_{|L} ∋ w ↦ (γw, w) ∈ i_{L}^{∧}(A). Therefore we can identify i_{L}^{∧}(A) with A_{|L} to obtain a transitive Lie algebroid on L. The bracket in Sec(A_{|L}) satisfies the condition

[[ξ|L, η|L]] = [[ξ, η]]|L, ξ, η ∈ Sec A.

We shall call A_{|L} the restriction of A to the leaf L.

The second important ”technical” notion is the inverse image f^{⋆}T of a representation T : A → A(𝔣) over f : (M′, F′) → (M, F); see [5]. For the inclusion i_{L}, we obtain in this way the restriction T_{|L} : A_{|L} → A(𝔣_{|L}) of T to L as a representation such that

T_{|L}(v)(ν|L) = T (v)(ν) for ν ∈ Sec 𝔣.

1.6 Representation of principal bundles on vector bundles

Denote by L𝔣 the GL(V)-principal bundle of all frames z : V ⟶^{≅}𝔣_{|x}, x ∈ M (V being the typical fibre of 𝔣). Let μ : G → GL(V) be a homomorphism of Lie groups.

Definition 1.6.1 By a μ-representation of a principal bundle P = P (M, G) on 𝔣 we mean [5] a μ-homomorphism H : P → L𝔣 between principal bundles. The homomorphism H after its differentiation gives a homomorphism dH : A(P) → A(L𝔣) of Lie algebroids [4], [7]; on the other hand, A(L𝔣) is canonically isomorphic to the Lie algebroid A(𝔣) via some isomorphism Φ_{𝔣} : A(L𝔣) → A(𝔣) defined by

Φ_{𝔣}([v])(ν) = u(∂_{v}(ν̃)),

where v ∈ T_{u}(L𝔣), u ∈ L𝔣, ν ∈ Sec 𝔣, whereas

ν̃ : L𝔣 → V, w ↦ w^{−1}(ν(πw)),

see [5]. The superposition H′ = Φ_{𝔣} ∘ dH : A(P) → A(𝔣) is called the differential of H.

Let H : P → L𝔣 be a μ-representation of P on 𝔣. A cross-section ν ∈ Sec 𝔣 is called H-invariant if there exists a vector v ∈ V such that H(z)(v) = ν_{π(z)} for all z ∈ P. Let (Sec 𝔣)_{I(H)} denote the space of all H-invariant cross-sections of 𝔣. The space (Sec 𝔣)_{I(H)} is isomorphic to the space of μ-invariant elements V_{I(μ)} via the isomorphism V_{I(μ)} → (Sec 𝔣)_{I(H)} given by v ↦ ν_{v} where ν_{v}(x) = F (z)(v) for all z ∈ P_{|x}, x ∈ M [5; 5.5.2].

The crucial role in the sequel of the theory is played by the following

Theorem 1.6.2 [5; 5.5.3] The spaces of invariant cross-sections (Sec 𝔣)_{I(H)} and (Sec 𝔣)_{I^{∘}(H′)} under a representation H : P → L𝔣 and its differential H′ : A(P) → A(𝔣) are related by (Sec 𝔣)_{I(H)} ⊂ (Sec 𝔣)_{I^{∘}(H′)}. If P is connected (nothing is assumed about the connectedness of G !), then

(Sec 𝔣)_{I(H)} = (Sec 𝔣)_{I^{∘}(H′)}. □

Let ggg be the LAB adjoint of A(P). By the adjoint representation of P we mean the (Ad_{G} : G → GL(V))-representation Ad_{P} : P → Lggg defined by Ad_{P} (z) = ẑ where ẑ : 𝔤 → ggg_{|π(𝔷)} is an isomorphism of Lie algebras defined by ẑ(v) = [(A_{z})_{⋆e}(v)], where A_{z} : G → P, a ↦ za, (𝔤 denotes here the right! Lie algebra of G). According to [5], we have

(Ad_{P})′ = ad_{A(P)}.

2 Partially invariant cross-sections

Let (A, [[·, ·]], γ) and 𝔣 be a transitive Lie algebroid and a vector bundle on a manifold M, respectively. Assume that F ⊂ T M is a C^{∞} constant dimensional involutive distribution and F – the foliation determined by F. We recall that A and F give rise to the regular Lie algebroid over (M, F) where we put A^{F} := γ^{−1}[F] ⊂ A; see [5; 1.1.3]. Its Atiyah sequence is

0 ⟶ ggg ↪ A^{F} ⟶^{γ^{F}} F ⟶ 0 ,

where ggg is the Lie algebra bundle adjoint of A, and γ^{F} := γ|A^{F}. Any representation T : A → A(𝔣) of A on 𝔣 [5; 2.1.1] restricts to the representation

T^{F} = T |A^{F} : A^{F} ⟶ A(𝔣)

of A^{F} on 𝔣. Any T^{F} –invariant cross-section ν ∈ Sec 𝔣 will be called a partially invariant cross-section with respect to T over F.

For any leaf L ⊂ M of the foliation F, we have the restriction A_{|L}^{F} of A^{F} and the inverse image

T_{|L}^{F} : A_{|L}^{F} ⟶ A(𝔣_{|L})

of T^{F} .

According to this, the following lemma is obvious.

Lemma 2.1 A cross-section ν ∈ Sec 𝔣 is T^{F} -invariant if and only if, for each leaf L of F, the restriction ν|L ∈ Sec 𝔣_{|L} is T_{|L}^{F} -invariant. □

Lemma 2.2 For finitely many F-basic functions f^{i} ∈ Ω_{b}^{∘}(M, F) and T -invariant cross-sections ν_{i} ∈ Sec 𝔣, ∑_{i} f^{i} · ν_{i} is a T^{F} -invariant cross-section, in other words,

Ω_{b}^{∘}(M, F) · (Sec 𝔣)_{I^{∘}(T)} ⊂ (Sec 𝔣)_{I^{∘}(T^{F})} .  (2.1)

Proof. For ξ ∈ Sec A^{F}, we obtain

L_{T^{F} ∘ ξ}(∑_{i} f^{i} · ν_{i}) = ∑_{i} f^{i} · L_{T^{F} ∘ ξ}(ν_{i}) + (γ∘ ξ)(f^{i}) · ν_{i} = 0

because L_{T^{F} ∘ ξ}(ν_{i}) = L_{T ∘ ξ}(ν_{i}) = 0 by the T -invariance of ν_{i}, and (γ∘ ξ)(f^{i}) = 0 by the fact that f^{i} ia basic and γ∘ ξ ∈ Sec F. □

Inclusion (2.1) can not always be replaced by the equality, which means that, in general, not every T^{F} -invariant cross-section is of the form ∑_{i} f^{i} · ν_{i} for F-basic functions f^{i} and T -invariant cross-sections ν_{i}, see example (2.4) below.

An important class of examples in which (2.1) is the equality is described later in Th.3.2.2.

Definition 2.3 Each T^{F} -invariant cross-section ν ∈ (Sec 𝔣)_{I^{∘}(T^{F})} not belonging to Ω_{b}^{∘}(M, F) · (Sec 𝔣)_{I^{∘}(T)} will be called singular. The characteristic class corresponding to any singular cross-section will be also called singular.

Example 2.4 Consider the Möbius band M with the foliation F by ”meridians”, see the Fig. 1.

Fig. 1

Let F denote the tangent bundle to F. Equip M with a flat Riemannian structure for which the fields e_{1} = \frac{∂}{∂x} and e_{2} = \frac{∂}{∂y} form an orthonormal base. The vector field e_{1} on M is not continuous at points of the segment AB.

Let P be the O(2, ℝ)-principal bundle of orthonormal frames of the tangent bundle T M. P is connected because M is not orientable [2] (therefore the structure Lie group O(2, ℝ) cannot be reduced to SO(2, ℝ)).

Consider the Atiyah sequence of the Lie algebroid (A(P), [[·, ·]], γ) of the principal bundle P:

0 ⟶ ggg ↪ A(P) ⟶^{γ} T M ⟶ 0 ,

where ggg ≅ End_{Sk}(T M, T M) is the vector bundle of skew-symmetric endomorphisms ( each element σ ∈ ggg_{|x} with respect to the base e_{1x}, e_{2x} has a matrix of the form \left[{0 c \atop −c 0}\right] for some real c).

Denote by A(ggg) the Lie algebroid of the vector bundle ggg. Consider the adjoint representation ad_{A(P)} : A(P) → A(ggg) and the one ad_{A(P)}^{F} : A(P)^{F} → A(ggg) determined by ad_{A(P)}.

For a real c ∈ ℝ, ˜ denotes the cross-section of ggg whose value at x ∈ M is an endomorphism T_{x}M ⟶ T_{x}M with the matrix \left[{0 c \atop −c 0}\right]. The cross-section c̃ is continuous except at points of the segment AB, and c̃ restricted to any leaf L of the foliation F, c̃|L, is:

1) smooth,

2) invariant with respect to the restricted representation ad(L) : A(P)_{|L}^{F} ⟶ A(ggg)_{|L} (equal to the adjoint representation of the Lie algebroid of the principal bundle P_{|L}).

It is evident to see the property 1) holds. As for 2), leaves of F are diffeomorphic to the segment (0, 1) ⊂ ℝ, therefore the transitive Lie algebroid A(P)_{|L}^{F} is trivial: A(P)_{|L}^{F} ≅ T I × Sk(2, ℝ), 𝕀 = (⊬, ⊮) ⊂ ℝ, see 1.1. In this identification, c̃ is a constant crosssection (0, \left[{0 c \atop −c 0}\right]). The equality [[ξ, c̃]] = 0 for any ξ ∈ Sec(T I × Sk(2, ℝ)) now follows trivially from the definition of the bracket in the Lie algebra Sec(T I × Sk(2, ℝ)) for the trivial Lie algebroid T I × Sk(2, ℝ). The property 2) can be also noticed in another way by using Proposition 5.5.2 in [5]: L is contractible, so P_{|L} has a reduction to SO(2, ℝ), say, P′_{L}. On the other hand, Sk(2, ℝ) is abelian, therefore each of its elements is invariant with respect to the adjoint representation of Sk(2, ℝ) [2], and c̃ corresponds (via the isomorphism described in Prop. 5.6.2 [5], applied to P′_{L}) to the invariant element \left[{0 c \atop −c 0}\right].

Define a cross-section Pf ∈ Secggg^{∗} by the formula Pf(\left[{0 c \atop −c 0}\right]) = c. Of course, Pf is continuous except at points of the segment AB. Besides, Pf has properties analogous to 1) and 2) above (we only need to change the representation ad_{A(P)} by its contragredient). Notice that Pf restricted to L corresponds to the Pfaffian [1] for the principal bundle P′_{L}.

The last step of our construction will be the smoothing of Pf which consists in multiplying it by a suitable basic function g. First, we notice that the space of leaves of F is the circle S^{1} with a base point x_{0} corresponding to the segment AB. Next, g is defined in such a way that it is C^{∞} and has only one zero at x_{0} and all the derivatives are also zero at x_{0}. Clearly, Pf · g is the sought-for partially invariant cross-section.

3 The tangential Chern-Weil homomorphism

3.1 The tangential Chern-Weil homomorphism of a transitive Lie algebroid

Definition 3.1.1 By the tangential Chern-Weil homomorphism of a transitive Lie algebroid A, over a foliated manifold (M, F), we mean the Chern-Weil homomorphism

h_{A^{F}} : I^{∘}(A^{F}) ⟶ H_{F} (M)

of the regular Lie algebroid A^{F}.

Clearly, h_{A^{F}} measures the nonexistence of flat connections in A^{F}.

For any leaf L of the foliation F, the Atiyah sequence of the transitive Lie algebroid A_{|L}^{F} is

0 ⟶ ggg_{|L} ↪ A_{|L}^{F} ⟶^{γ_{|L}} T L ⟶ 0 .

Without any difficulties we can verify the following Proposition 3.1.2 The Chern-Weil homomorphisms h_{A} of A, h_{A^{F}} of A^{F} and h_{A_{|L}^{F}} of A_{|L}^{F} are connected with one another via the commuting diagram

I^{∘}(A)  ⟶^{h_{A}} H(M)
↓_{1·Γ}^{Γ} ↓  ↓ ↓_{α^{F}}^{α}

Ω_{b}^{∘}(M, F)·I^{∘}(A) ⟶^{i} I^{∘}(A^{F}) ⟶^{h_{A^{F}}} H_{F} (M)
↓_{Γ|L}^{Γ} ↓  ↓ ↓_{θ|L}^{θ}

I^{∘}(A_{|L}^{F}) ⟶^{h_{A{F \atop |L}}} H_{dR}(L)

where α^{F} is the tangential cohomology class whose representantive comes from some representantive of α by restricting it to vectors from the distribution F. □

Notice that the triviality of h_{A} implies the same assertion for h_{A^{F}} ∘ i :

h_{A^{F}} ∘ i(∑_{i} f^{i} · Γ_{i}) = ∑_{i} f^{i} · (h_{A}(Γ_{i})) = 0.

3.2 The tangential Chern-Weil homomorphism of a principal bundle

Let P be a G-principal bundle on M, whereas F ⊂ T M and F are as above.

Definition 3.2.1 By the tangential Chern-Weil homomorphism of P over (M, F) we mean the Chern-Weil homomorphism h_{A(P)^{F}} of the regular Lie algebroid A(P)^{F}.

Let 𝔤 be the right Lie algebra of G and (⋁ 𝔤^{⋆})_{ℑ(𝔊)} — the algebra of G-invariant elements [2].

Theorem 3.2.2 Let G_{∘} be the connected component containing the unit of G. If each G_{∘}-invariant element of ⋁ 𝔤^{⋆} is G-invariant, then the domain of the homomorphism h_{A(P)^{F}} is equal to Ω_{b}^{∘}(M, F)·I^{∘}(A(P)) (≅ Ω_{b}^{∘}(M, F)·(⋁ 𝔤^{⋆})_{ℑ(𝔊)} when P is connected).

Remark 3.2.3 The assumptions of the above theorem are satisfied, for example, when G is connected or when G = GL(V).

Proof of Theorem 3.2.2. Since the representation ad_{A(P)} is the differential of the representation Ad_{P} and the same holds for the symmetric powers of their contragredients (see Th. 5.4.3 in [5]), the theorem clearly follows from the following Proposition. □ Proposition 3.2.4 Let A = A(P) for a G-principal bundle P on a manifold M, and let 𝔣 be a vector bundle on M with a typical fibre V. Assume that T : A(P) → A(𝔣) is the differential of some (μ : G → GL(V))-representation H : P → L(𝔣) of P on 𝔣 (see 1.6). If each (μ|G_{∘} : G_{∘} → GL(V))-invariant vector v ∈ V is μ-invariant, then, for any involutive distribution F on M, each T^{F} -invariant cross-section of 𝔣 is of the form of a finite sum ∑_{i} f^{i} ·Ψ_{i} for F-basic functions f^{i} and T -invariant cross-sections Ψ_{i}.

Proof. According to Propositions 5.5.2–3 in [5] we have

V_{I} ≅ (Sec 𝔣)_{I(H)} ⊂ (Sec 𝔣)_{I^{∘}(T)}

where V_{I} is the space of μ-invariant vectors, whereas (Sec 𝔣)_{I(H)} and (Sec 𝔣)_{I^{∘}(T)} are the spaces of invariant cross-sections with respect to H and T, respectively.

Let ν ∈ Sec 𝔣 be T^{F} -invariant. In particular, ν_{x} ∈ 𝔣_{|x} is (T^{F})_{|x}^{+} (= T_{|x}^{+})-invariant (for the index ”+”, see section 1.1). We recall, see section 1.2 in [5], that the LAB adjoint of A(𝔣) is isomorphic to End 𝔣. Via this isomorphism, the following diagram

𝔤 ⟶^{dμ} End(V)
ẑ↓  ↓ ↓_{H(z)∘a∘H(z)^{−1}}^{a}

ggg_{|x} ⟶^{T_{|x}^{+}} End (𝔣_{|x})

commutes for each z ∈ P (ẑ is defined in section 1.6). This easily implies that, for z ∈ P, the vector w_{x} := H(z)^{−1}(ν_{x}) ∈ V is dμ-invariant. By the assumption with G, w_{x} is μ-invariant [2]. Let (w_{1}, …, w_{k}) be a base of the space V_{I} of μ-invariant vectors. Then there exists real numbers f^{1}(x), …, f^{k}(x) such that w_{x} = ∑_{i} f^{i}(x) · w_{i}; on the other hand, we have the equality ν = ∑_{i} f^{i} · ν_{w_{i}} where ν_{w_{i}} are H-invariant (therefore T -invariant) cross-sections defined in 1.6. The linear independence of ν_{w_{i}} proves the smoothness of f^{i}. Finally, we prove that f^{i} ∈ Ω_{b}^{∘}(M, F). Let X ∈ Sec F. There exists ξ ∈ Sec A(P)^{F} such that γ ∘ ξ = X. Therefore we obtain

0 = L_{T^{F} ∘ξ}(∑_{i} f^{i} · ν_{w_{i}}) = ∑_{i}(f^{ı} · L_{T ∘ξ}(ν_{w_{i}}) + (γ ∘ ξ)(f^{i}) · ν_{w_{i}}) = ∑_{i} X(f^{i}) · ν_{w_{i}},

which implies X(f^{i}) = 0. The free choice of X proves that f^{i} are basic. □

3.3 The tangential Chern-Weil homomorphism of a vector bundle

Let 𝔣 ⟶^{p} M be a vector bundle with typical fibre V and let G ⊂ GL(V) be a Lie subgroup of GL(V). Assume that A is any maximal family of distinguished local trivializations ψ : U_{ψ} × V ⟶ p^{−1}[U_{ψ}], U_{ψ} being an open subset of M and ⋃ U_{ψ} = M such that the transition functions have values in G. The pair (𝔣, A), called a vector bundle with the structure Lie group G (or in the sequel, briefly, a G-vector bundle), determines

a) the G-principal bundle L(𝔣, A) of distinguished frames

L(𝔣, A) = {ψ_{|x} : V ⟶ 𝔣_{|x}; ψ ∈ A, x ∈ U_{ψ}},

b) the transitive Lie algebroid denoted by A(𝔣, A) (being a Lie subalgebroid of A(𝔣)),

putting

A(𝔣, A) = Φ_{𝔣}[A(L(𝔣, A))]

(we recall that Φ_{𝔣} : A(L𝔣) ⟶ A(𝔣) is the canonical isomorphism of transitive Lie algebroids, see section 1.6).

Definition 3.3.1 By the tangential Chern-Weil homomorphism of a G-vector bundle (𝔣, A), over a foliated manifold (M, F), we mean the Chern-Weil homomorphism

h_{A(𝔣,A)^{F}} : I^{∘}(A(𝔣, A)^{F}) ⟶ H_{F}(M)

of the regular Lie algebroid A(𝔣, A)^{F}.

Now, we describe elements of the Lie algebroids A(𝔣, A) in terms of the vector bundle (𝔣, A).

Theorem 3.3.2 Let l ∈ A(𝔣)_{|x} , x ∈ M, then l ∈ A(𝔣, A)_{|x} if and only if, for any ψ ∈ A such that x ∈ U_{ψ}, the endomorphism

V ∋ u ⟼ ψ_{|x}^{−1}(l(ψ(·, u))) ∈ V

belongs to the Lie algebra of G.

We begin with the following lemma.

Lemma 3.3.3 For an arbitrary local trivialization ψ : U × V ⟶ p^{−1}[U] of a vector bundle 𝔣, the following diagram

A(L𝔣)  ⟶^{Φ_{𝔣}}  A(𝔣)
ψ̂^{A}↑  ↑ψ̄
T (U) × T_{id}(GL(V)) ⟶^{id×ρ_{V}} T (U) × End(V)

is commutative when

1) ψ̂^{A} is a local trivialization determined by ψ̂ : U ×GL(V) → L𝔣, (x, a) ↦ ψ(x, ·)∘ a, via the formula (see [4; 1.1]) ψ̂^{A}(v, w) = [(ψ̂)_{⋆(x,e)}(v, w)], 2) ψ̄ is described in Th. 1.3.1,

3) ρ_{V} : T_{id}(GL(V)) → End (V) is an isomorphism of Lie algebras (T_{id}(GL(V)) is meant as a right Lie algebra), defined by [5; 5.2.1]

ρ_{V} (w) : V ⟶ V, u ↦ ∂_{v}(ũ),

where ũ : GL(V) → V, a ↦ a^{−1}(u).

Proof. For v ∈ T_{x}U, w ∈ T_{id}(GL(V)) and ν ∈ Sec 𝔣, we obtain (see 1.6)

Φ_{𝔣}∘ψ̂^{A}(v, w)(ν) = Φ_{𝔣}([ψ̂_{∗}(v, w)])(ν)
= ψ(x, e)(∂_{ψ̂_{∗}(v,w)}(ν̃))
= ψ_{|x}(∂_{(v,w)}(ν̃∘ψ̂))
= ψ_{|x}(∂_{v}(ν̃∘ψ̂(·, e)) + ∂_{w}(ν̃∘ψ̂(x, ·)))
= ψ_{|x}(∂_{v}(ν_{ψ})) + ψ_{|x}(ρ_{V} (w)(ν_{ψ}(x)))
= ψ̄(v, ρ_{V} (w))(ν)
= ψ̄(id × ρ_{V})(v, w)(ν). □

Proof of Theorem 3.3.2. Suppose that x ∈ M and take an arbitrary local trivialization ψ ∈ A such that x ∈ U_{ψ}. Clearly, the restriction ψ̂_{G} of ψ̂ to U × G,

ψ̂_{G} : U × G ⟶ L(𝔣, A), (y, a) ⟼ ψ(y, ·)∘a ,

is a local trivialization of the G-principal bundle L(𝔣, A). Passing to algebroids, we have the following diagram:

A(L(𝔣, A)) ↪  A(L𝔣)  ⟶^{Φ_{𝔣}}  A(𝔣)
↑ ψ̂_{G}^{A}  ↑ ψ̂^{A}  ↑ ψ̄
T U × T_{id}G ↪ T U × T_{id}(GL(V)) ⟶^{id×ρ_{V}} T U × End(V) .

Understanding GL(V) as an open subset of End(V), we have the canonical isomorphism T_{id}(GL(V)) ≅ End(V); then ϱ_{V} = −id (see 5.2.1 in [5]). Let 𝔤 ⊂ End(𝔙) be the Lie algebra of G. Clearly, ϱ_{V} [T_{id}G] = 𝔤, therefore, according to the above diagram, the restriction ψ̄_{𝔤} of ψ̄ to T U × 𝔤 has values in A(𝔣, A) and ψ̄_{𝔤} : T U × 𝔤 → 𝔄(𝔣, A) is a local trivialization of the Lie algebroid A(𝔣, A).

Take l ∈ A(𝔣)_{|x}. From the above we obtain:

l ∈ A(𝔣, A)_{|x} ⟺ ψ̄_{|x}^{−1}(l) ∈ T U × 𝔤.

To prove the ” ⟹ ” part of our theorem, assume that l ∈ A(𝔣, A)_{|x} and write l = ψ̄_{|x}(q(l), a) for a ∈ 𝔤 (q(l) is the anchor of l, see 1.3) and notice that we must show the relation a(u) = ψ_{|x}^{−1}(l(ψ(·, u))), i.e. equivalently, that ψ_{|x}(a(u)) = l(ψ(·, u)). But from the fact ν_{ψ(·,u)} ≡ u, we obtain

ψ_{|x}(a(u)) = ψ_{|x}(∂_{q(l)}(ν_{ψ(·,u)}) + a(ν_{ψ(·,u)}(x)))
= ψ̄(q(l), a)(ψ(·, u))
= l(ψ(·, u)).

To prove the ” ⟸ ” part, assume that a := (u ↦ ψ_{|x}^{−1}(l(ψ(·, u)))) belongs to 𝔤. It only remains to notice that l = ψ̄_{|x}(q(l), a). However, l and ψ̄_{|x}(q(l), a) are 𝔣vectors with the same anchors, so, to show their identity, it is sufficient to check their behaviour on the cross-sections generating the module Sec 𝔣 near x, for example, on ψ(·, u), u ∈ V:

ψ̄_{|x}(q(l), a)(ψ(·, u)) = ψ_{|x}(∂_{q(l)}(ψ(·, u)_{ψ}) + ψ_{|x}^{−1}(l(ψ(·, ψ(·, u)_{ψ}(x))))
= l(ψ(·, u)). □

If ∇ is a covariant derivative in 𝔣, then, for v ∈ T_{x}M, x ∈ M, the linear mapping

∇_{v} : Sec 𝔣 ⟶ 𝔣_{|x}, ν ⟼ ∇_{v}ν,

is clearly an 𝔣-vector, i.e. ∇_{v} ∈ A(𝔣)_{|x}. Besides, the mappimg

T M ∋ v ⟼ ∇_{v} ∈ A(𝔣)

is a connection in A(𝔣). Conversely, any connection λ : T M → A(𝔣) determines a covariant derivative ∇ in 𝔣 by the formula

∇_{v}(ν) = λ(v)(ν), v ∈ T M, ν ∈ Sec 𝔣.

Lemma 3.3.4 If ∇ is the covariant derivative in 𝔣 corresponding to a connection λ in A(𝔣), then the curvature tensor R ∈ Ω^{2}(M; End 𝔣) of ∇ is equal to the curvature tensor Ω of λ multiplied by -1.

Proof. For X, Y ∈ X (M) and ν ∈ Sec 𝔣, we have (see 1.3):

Ω(X, Y)(ν) = L_{λ∘[X,Y]}(ν) − L_{[[λ∘X,λ∘Y]]}(ν)
= ∇_{[X,Y]}(ν) − [∇_{X}, ∇_{Y}](ν)
= −R_{X,Y} (ν). □

Corollary 3.3.5 A connection λ in A(𝔣) is flat over an involutive distribution F ⊂ T M if and only if the corresponding covariant derivative in 𝔣 is flat over F. □

Definition 3.3.6 A covariant derivative ∇ in 𝔣 will be called a covariant derivative in the G-vector bundle (𝔣, A) if the corresponding connection ∇ in A(𝔣) has values in A(𝔣, A).

Clearly, the correspondence (described above) between connections in A(𝔣, A) and covariant derivatives in (𝔣, A) is one-to-one.

According to Th. 3.3.2, ∇ is a covariant derivative in (𝔣, A) if and only if, for any ψ ∈ A, x ∈ U_{ψ} and v ∈ T_{x}M, the endomorphism

V ∋ u ⟼ ψ_{|x}^{−1}(∇_{v}(ψ(·, u))) ∈ V

belongs to the Lie algebra 𝔤 of G.

Remark 3.3.7 Important examples of a reduction of the structure Lie group GL(V) in 𝔣 can be obtained via the so-called Σ-bundles [2; Ch.VIII]. Let (𝔣, Σ_{𝔣}) be an arbitrary Σ-bundle, Σ_{𝔣} being a finite ordered set of cross-sections ν_{i} of 𝔣^{p_{i},q_{i}} (:= ⊗^{p_{i}}𝔣^{⋆} ⨂ ⊗^{q_{i}}𝔣), i ≤ m, subject to the following condition:

there is a finite ordered system Σ_{V} = {v_{1}, …, v_{m}} of tensors v_{i} ∈ V^{p_{i},q_{i}} (V is the typical fibre of 𝔣) and there is a system of local trivializations A of 𝔣 such that, for each ψ ∈ A, ψ_{|x}^{p_{i},q_{i}}(v_{i}) = ν_{i}(x), i ≤ m, x ∈ U_{ψ}.

The pair (𝔣, A) is then a vector bundle with a reduction to the closed Lie subgroup G ⊂ GL(V) consisting of those and only those linear isomorphisms φ : V → V for which φ^{p_{i},q_{i}}(v_{i}) = v_{i}, i ≤ m. The Lie algebra 𝔤 of G is the subalgebra of End(V) consisting of the linear transformations φ of V which satisfy θ_{φ}^{p_{i},q_{i}}(v_{i}) = 0, i ≤ m, where θ^{p_{i},q_{i}} is the canonical representation of the Lie algebra End(V) on V^{p_{i},q_{i}} [2]. Denote also by θ^{p_{i},q_{i}} the canonical representation of the Lie algebroid A(𝔣) on 𝔣^{p_{i},q_{i}}, generated by the identical one id_{A(𝔣)} : A(𝔣) → A(𝔣); see [5; 2.2].

By the above, there are no essential difficulties to prove

Proposition 3.3.8 Let l ∈ A(𝔣)_{|x}, x ∈ M; then l ∈ A(𝔣, Σ_{𝔣})_{|x} if and only if θ^{p_{i},q_{i}}(l)(ν_{i}) = 0, i ≤ m. □

A covariant derivative ∇ in 𝔣 detrmines, in the standard way, the covariant derivative ∇^{p_{i},q_{i}} in 𝔣^{p_{i},q_{i}}. Clearly, we have ∇^{p_{i},q_{i}} = θ^{p_{i},q_{i}} ∘ ∇. According to this and the last proposition, we obtain

Corollary 3.3.9 A covariant derivative ∇ is in (𝔣, Σ_{𝔣}) if and only if ∇^{p_{i},q_{i}}(ν_{i}) = 0, i ≤ m, i.e. if and only if ∇ is a Σ-connection. □

As an example consider a Riemannian bundle (𝔣, {G}), G being a Riemannian tensor in 𝔣. In this case, covariant derivatives in (𝔣, {G}) are simply Riemannian connections.

Now on the base M let us define a foliation F having F ⊂ T M as its tangent bundle. If λ is a connection in the G-vector bundle (𝔣, A), then the operator ∇ defined by ∇_{v}(ν) = λ(v)(ν), v ∈ F, ν ∈ Sec 𝔣, is a partial covariant derivative in 𝔣 over F in the sense of [3]. Conversely, a partial covariant derivative in 𝔣 over F such that ∇_{v} ∈ A(𝔣, A) for v ∈ F (called a partial covariant derivative in (𝔣, A) over F) determines a connection F ∋ v ↦ ∇_{v} ∈ A(𝔣, A) in A(𝔣, A)^{F}. This correspondence is one-to-one. Clearly, in the case of a Σ-bundle (𝔣, Σ_{𝔣}), a partial covariant derivative ∇ in 𝔣 over F is in the bundle (𝔣, Σ_{𝔣}) if and only if ∇ν = 0 for ν ∈ Σ_{𝔣}.

Now, pass to the investigation of the Chern-Weil homomorphism of the regular Lie algebroid A(𝔣, A)^{F}. According to the above, its nontriviality means that in (𝔣, A) there is no flat partial covariant derivative over F.

3.4 Open problem

— Find some G-vector bundle (𝔣, A) over a foliated manifold, possessing nontrivial singular characteristic classes.

Let Pont(A) := Im(h_{A}) be the Pontryagin algebra of a regular Lie algebroid A. Consider a nonorientable Riemannian vector bundle 𝔣 of rank 2m and a connected O(2m; ℝ)-principal bundle P of orthonormal frames of 𝔣, and the transitive Lie algebroid A = A(P). We have Pont^{2m}(P) = Pont^{2m}(A) = 0 (and, of course, Pont^{k}(P) = 0 for k > 2m).

Conjecture 3.4.1 There exists an example of a nonorientable Riemannian vector bundle 𝔣 and an involutive distribution F with orientable leaves on the base M of 𝔣, for which

Pont^{2m}(A^{F}) ≠ 0.

(Stronger conjecture: 𝔣 can be taken as the tangent bundle T M to some manifold M).

Singular partial invariant cross-sections from the domain of h^{2m}(A^{F}) can be obtained by ”gluing” and ”smoothing” the cross-sections (defined on leaves) coming from the Pfaffians (as in Example 2.1).

References

[1] J. L. Dupont, Curvature and characteristic classes, Lecture Notes in Mathematics 640, Springer-Verlag 1978.

[2] W. Greub, and S. Halperin, and R. Vanstone, Connections, Curvature, and Cohomology, Vol II, Academic Press, New York and London, 1973.

[3] F. Kamber, and Ph. Tondeur, Foliated Bundles and Characteristic Classes, Lectures Notes in Mathematics 493, Springer-Verlag 1975.

[4] J. Kubarski, Lie algebroid of a principal fibre bundle, Publ. Dep. Math. Universite de Lyon 1, 1/A, 1989.

[5] J. Kubarski, The Chern-Weil homomorphism of regular Lie algebroids, Publ. Dep. Math. Universite de Lyon 1, (to appear).

[6] J. Kubarski, Characteristic homomorphisms of flat and of partially flat regular Lie algebroids, Publ. Dep. Math. Universite de Lyon 1, (to appear).

[7] K. Mackenzie, Lie groupoids and Lie algebroids in differential Geometry, London Mathematical Society Lecture Note Series 124, Cambridge, 1987.

[8] C. C. Moore, and C. Schochet, Global Analysis on Foliated Spaces, Mathematical Sciences Research Institute publications; 9, 1988, Springer-Verlag New-York.