DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 309–319

SMOOTH CONNECTIONS AND HORIZONTAL DISTRIBUTIONS

ON MANIFOLDS OVER LOCAL ALGEBRAS

V.V.Shurygin

Abstract. On manifolds over local algebras special distributions complementary to leaves of the canonical foliation and smooth connections are considered. The main purpose is to study obstructions to existence of such distributions in terms of the Atiyah classes of foliated and smooth principal bundles. Besides that the notion of holonomy of a manifold over local algebra is introduced, and it is shown that existence of complementary distributions under consideration on a manifold over local algebra depends on its holonomy.

§1. Introduction

As it was shown by M.Atiyah [1], in order that complex analytic principal bundle admit a complex analytic connection it is necessary and sufficient that a certain cohomology class, called the Atiyah class, vanish. The present paper is devoted to the problem of existence of smooth connections and special horizontal distributions, which become connections in the case of A.Weil’s bundles of infinitely near points [2] (see also [3],[4]), on manifolds over local algebras. There are canonical foliations on manifolds over local algebras, and obstructions to existence of smooth connections and special horizontal distributions are related with obstructions to existence of projectable connections in foliated principle bundles defined by P.Molino [5] and by F.W.Kamber and P.Tondeur [6].

We adopt the following notation (see [7]):

ℝ(N, q) is the algebra of truncated real polynomials of degree q in N variables, 𝔸 = ℝ(N, q)/𝕀 a local algebra of width N and height q in the sense of A.Weil

(see [2], [3], [4]),

𝔸^{∘} the maximal ideal of 𝔸,

𝔸^{n} the 𝔸-module of n-tuples of elements of 𝔸, 𝔸^{∘}^{n} the submodule of 𝔸^{n} consisting of n-tuples of elements of 𝔸^{∘}, π_{0}^{q} : 𝔸 → ℝ and π_{0}^{q} : 𝔸^{n} → ℝ^{n} the natural epimorphisms (for simplicity we use

1991 Mathematics Subject Classification. 53C05, 53C12, 57R30, 58A20.

Key words and phrases. Foliation, manifold over algebra, d_{F} -cohomology, (X, G)-foliation.

The research described in this publication was made possible in part by Grant N JGX100 from the International Science Foundation and Russian Government.

The paper is in final form and no version of it will be submitted for publication elsewhere.

the same symbol π_{0}^{q}).

The module 𝔸^{n} (and the algebra 𝔸, in particular) can be represented as the

direct sum of vector spaces 𝔸^{n} = ℝ^{n} ⊕ 𝔸^{∘}^{n}, and so every X ∈ 𝔸^{n} can be uniquely

represented in the form X = x + X^{∘} where x ∈ ℝ^{n} and X^{∘} ∈ 𝔸^{∘}^{n}. The submersion

π_{0}^{q} : 𝔸^{n} → ℝ^{n} defines a foliation on 𝔸^{n} whose leaves are cosets of 𝔸^{∘}^{n}. We shall

call it the 𝔸^{∘}-foliation.

A smooth mapping

Φ : U ⊂ 𝔸^{n} → U′ ⊂ 𝔸^{k}  (1)

is called 𝔸-smooth if the tangent mapping T_{X}Φ is 𝔸-linear for any X ∈ U. If U is

a simple open subset [5] for the 𝔸^{∘}-foliation, an 𝔸-smooth mapping is of the form

Y^{i′} = ϕ^{i′}(x^{i}) + ∑_{|p|=1}^{q} \frac{1}{p!} \frac{D^{p}ϕ^{i′}}{Dx^{p}} X^{∘}^{p}  (2)

where i = 1, …, n, i′ = 1, …, k, p = (p_{1} …, p_{n}) is a multiindex of length n, and ϕ^{i′} : π_{0}^{q}(U) → 𝔸 are 𝔸-valued smooth functions. In this case the mapping (1) is uniquely extended to the 𝔸-smooth mapping

Φ : (π_{0}^{q})^{−1}(π_{0}^{q}(U)) → (π_{0}^{q})^{−1}(π_{0}^{q}(U′))  (3)

given by the same equations (2). Moreover, if the mapping (1) is an 𝔸-diffeomorphism (for n = k), the mapping (3) is an 𝔸-diffeomorphism too. If the domain U of a mapping (1) is not simple, it can be given by the formula (2) where ϕ^{i′} : U → 𝔸

are basic (projectable) functions for 𝔸^{∘}-foliation.

Every 𝔸-diffeomorphism Φ of the form (3) maps leaves of the 𝔸^{∘}-foliation into

leaves. In particular, if π_{0}^{q}(Φ(0)) = 0, the restriction Φ^{∘} = Φ|𝔸^{∘}^{n} : 𝔸^{∘}^{n} → 𝔸^{∘}^{n} is of the form

Y^{∘} ^{i} = ∑_{|p|=0}^{q} ϕ_{p}^{i}X^{∘}^{p}  (4)

where ϕ_{0}^{i} ∈ 𝔸^{∘}, ϕ_{p}^{i} ∈ 𝔸.

Let D_{n}(𝔸) denote the set of all diffeomorphisms of the form (4). D_{n}(𝔸) is a Lie group, the relations ϕ_{0}^{i} = 0, ϕ_{p}^{i} ∈ ℝ determine a Lie subgroup in D_{n}(𝔸) which is isomorphic to the differential group G_{n}^{q} (see [4]). D_{n}(𝔸) is the semidirect product

D_{n}(𝔸) = G_{n}^{q} × D^{∘}_{n}(𝔸) where the invariant subgroup D^{∘}_{n}(𝔸) is defined by the relations π_{0}^{q}(ϕ_{p}^{i}) = 0 for |p| > 1, π_{0}^{q}(ϕ_{j}^{i}) = δ_{j}^{i} [8]. We denote by τ : D_{n}(𝔸) → G_{n}^{q} the natural epimorphism, and by 𝔡_{n}(𝔸) the Lie algebra of D_{n}(𝔸).

By an n-dimensional smooth manifold over 𝔸 (𝔸-smooth manifold) we shall mean (see [9], [7]) a smooth manifold M^{𝔸} with a maximal atlas of charts h_{α} : U_{α} ⊂ M^{𝔸} → U′_{α} ⊂ 𝔸^{n} with 𝔸-smooth coordinate changes h_{α} ∘ h_{β}^{−1}. Such charts are called 𝔸-charts and an atlas is called an 𝔸-atlas. The bundle J^{𝔸}M of infinitely near points of 𝔸-type in the sense of A.Weil [2] (𝔸-jets [4], [7]) is an example of n-dimensional 𝔸-smooth manifold [3].

Let

X^{i′} = Φ^{i′}(X^{i})  (5)

be an 𝔸-smooth coordinate change on an 𝔸-smooth manifold M^{𝔸}. From (2) it follows that the real parts x^{i′} of the coordinates X^{i′} locally depend only on the real parts x^{i} of the coordinates X^{i}, and so the equations x^{i} = const determine a

foliation on M^{𝔸}. This foliation is called the canonical 𝔸^{∘}-foliation. Note that each submodule of 𝔸^{n} invariant under 𝔸-linear transformations generates a foliation on M^{𝔸} [7], and so M^{𝔸} is a multifoliated manifold.

When we fix x^{i} in (5), we obtain coordinate transformations on leaves of the

canonical 𝔸^{∘}-foliation. Locally these transformations are of the form (4), therefore each leaf of M^{𝔸} carries a structure of (X, G)-manifold in the sense of Thurston

(see [10]) for X = 𝔸^{∘}^{n} and G = D_{n}(𝔸), and M^{𝔸} carries a structure of tangential

(𝔸^{∘}^{n}, D_{n}(𝔸))-foliation (see [11], [12] about tangential (X, G)-foliations).

§2. Holonomy of a manifold over local algebra

Consider now an 𝔸-smooth manifold with an 𝔸-atlas {h_{α} : U_{α} ⊂ M^{𝔸} → U′_{α} ⊂ 𝔸^{n}}_{α∈A}. One can define the following two fiber bundles over M^{𝔸} with standard

fiber 𝔸^{∘}^{n} and structure groups D_{n}(𝔸) and G_{n}^{q} respectively:

1) π : O^{V} M^{𝔸} → M^{𝔸} with the atlas {H_{α} : π^{−1}(U_{α}) → U′_{α} × 𝔸^{∘}^{n}}_{α∈A} whose

transition functions H_{α} ∘ H_{β}^{−1} are of the form

\{X_{β}^{i}, X^{∗}_{β}^{i}\} ↦ \{X_{α}^{i} = ϕ_{αβ}^{i} + ∑_{|p|=1}^{q} \frac{1}{p!} \frac{D^{p}ϕ_{αβ}^{i}}{Dx_{β}^{p}} X^{∘}_{β}^{p}, X^{∗}_{α}^{i} = ϕ^{∘}_{αβ}^{i} + ∑_{|p|=1}^{q} \frac{1}{p!} \frac{D^{p}ϕ_{αβ}^{i}}{Dx_{β}^{p}} X^{∗}_{β}^{p}\}.

2) π̃ : Õ^{V} M^{𝔸} → M^{𝔸} with the atlas {H̃_{α} : π̃^{−1}(U_{α}) → U′_{α} × 𝔸^{∘}^{n}}_{α∈A} whose

transition functions H̃_{α} ∘ H̃_{β}^{−1} are of the form

\{X_{β}^{i}, X̂_{β}^{i}\} ↦ \{X_{α}^{i} = ϕ_{αβ}^{i} + ∑_{|p|=1}^{q} \frac{1}{p!} \frac{D^{p}ϕ_{αβ}^{i}}{Dx_{β}^{p}} X^{∘}_{β}^{p}, X̂_{α}^{i} = ∑_{|p|=1}^{q} \frac{1}{p!} \frac{D^{p}ϕ̄_{αβ}^{i}}{Dx_{β}^{p}} X̂_{β}^{p}\}

where ϕ_{αβ} = h_{α} ∘ h_{β}^{−1}, ϕ̄_{αβ} = π_{0}^{q} ∘ ϕ_{αβ}. The charts H_{α} and H̃_{α} will be called the charts corresponding to h_{α}.

The bundle Õ^{V} M^{𝔸} can be identified with the bundle J_{tr}^{𝔸}M^{𝔸} of transversal 𝔸-

jets over M^{𝔸} with respect to the canonical 𝔸^{∘}-foliation on M^{𝔸}. In the case when M^{𝔸} is complete, that is all leaves of the canonical 𝔸^{∘}-foliation are complete, the bundle O^{V} M^{𝔸} can be realized as the union of fundamental groupoids of leaves (see [12]). By P^{V} M^{𝔸} we shall denote the principal D_{n}(𝔸)-bundle associated with O^{V} M^{𝔸}. P^{V} M^{𝔸} can be realized as the set of germs of local 𝔸-affine mappings from

𝔸^{∘}^{n} into leaves of the canonical 𝔸^{∘}-foliation. The bundle J_{tr}M^{𝔸} is associated with the bundle B_{tr}^{q}M^{𝔸} of transversal q-frames.

Let L be a leaf of the canonical 𝔸^{∘}-foliation through X_{0} ∈ M^{𝔸}, γ : [0, 1] → L a continuous loop based at X_{0}, and (U, h) a fixed 𝔸-chart on M^{𝔸} with h(X_{0}) = 0 and U being simple open subset. Consider a collection of 𝔸-charts {U_{α_{k}}, h_{α_{k}}}, k = 1, …, s, on M^{𝔸} satisfying the following conditions: U_{α_{k}} is simple open set,

U_{α_{1}} = U_{α_{s}} = U, h_{α_{k}}(U_{α_{k}} ∩ L) ⊂ 𝔸^{∘}^{n}, and there is a subdivision t_{0} = 0 < t_{1} < ⋯ < t_{s} = 1 of [0, 1] such that γ([t_{k}, t_{k+1}]) ⊂ U_{k+1}. Each 𝔸-smooth coordinate change Φ_{k} = h_{α_{k}} ∘ h_{α_{k+1}}^{−1} restricted on a simple neiborhood of the point γ(t_{k}) can be extended to an 𝔸-diffeomorphism Φ_{k} of the form (3), therefore on a suitable domain (π_{0}^{q})^{−1}(V), where V is an open neighborhood of 0 ∈ ℝ^{n}, we can take the composition Φ_{0} ∘ Φ_{1} ∘ ⋯ ∘ Φ_{s}. The germ α(γ) of this composition is defined

along the entire submodule 𝔸^{∘}^{n} and depends only on a homotopy class of γ, and if γ_{3} = γ_{2} ∗ γ_{1}, then α(γ_{3}) = α(γ_{2}) ∘ α(γ_{1}). Thus we obtain a homomorphism

α : π_{1}(L, X_{0}) ∋ [γ] ↦ α(γ) ∈ Diff_{𝔸}(𝔸^{n}, 𝔸^{∘}^{n})

where Diff_{𝔸}(𝔸^{n}, 𝔸^{∘}^{n}) is the group of germs of 𝔸-smooth diffeomorphisms of the form (π_{0}^{q})^{−1}(V) ⊂ 𝔸^{n} → (π_{0}^{q})^{−1}(V′) ⊂ 𝔸^{n}, and V and V′ are neighborhoods of 0 ∈ ℝ^{n}.

We shall call α the holonomy representation of M^{𝔸} at X_{0}. The image of α will be called the holonomy group of M^{𝔸} at X_{0}.

Besides α the following three holonomy representations are associated with a

leaf L of the canonical 𝔸^{∘}-foliation: α_{tr}, the holonomy representation of L at X_{0} in

the sense of the theory of foliations [5], α_{l}, the holonomy of L as an (𝔸^{∘}^{n}, D_{n}(𝔸))- manifold ([10], §2.2), and ᾱ_{l} = τ ∘ α_{l}, which will be called the G_{n}^{q}-holonomy representation of L.

Theorem 1. The holonomy representations associated with M^{𝔸} are related by the equalities

α_{tr} = t ∘ α, α_{l} = r ∘ α, ᾱ_{l} = j^{q} ∘ α_{tr}  (6)

where t maps a germ Φ of 𝔸-smooth diffeomorphism into the germ π_{0}^{q} ∘ Φ|(ℝ^{n}, 0),

r is the restriction of germs to the leaf 𝔸^{∘}^{n} ⊂ 𝔸^{n},and j^{q} assigns to a germ its q-jet.

Proof. The relations (6) follow from the structure of equations (5), which provides the commutativity of the following diagram

Diff_{𝔸}(𝔸^{n}, 𝔸^{∘}^{n}) ⟶^{t} Diff(ℝ^{n}, 0)
r↓  ↓^{j^{q}}
D_{n}(𝔸)  ⟶^{τ}  G_{n}^{q}

Note. In the case of algebra ℝ(ε) a leaf L of the canonical ℝ^{∘}(ε)-foliation carries a structure of affine manifold and ᾱ_{l} is the linear holonomy representation [13] of L as an affine manifold. Relations similar to ᾱ_{l} = j^{q} ∘ α_{tr} have been found by T.Inaba [11] for Lagrangian and Legendrian foliations.

The holonomy representation α of M^{𝔸} at X_{0} depends on a choice of fixed 𝔸-chart (U, h). An interpretation of this holonomy independent of such a choice can be given in terms of the bundles O^{V} M and P^{V} M. Let us assign to a germ

Φ ∈ Diff_{𝔸}(𝔸^{n}, 𝔸^{∘}^{n}) written in coordinates in the form

Y^{i} = ϕ^{i}(x^{j}) + ∑_{|p|=1}^{q} \frac{1}{p!} \frac{D^{p}ϕ^{i}}{Dx^{p}} X^{∘}^{p}

the germ of projectable diffeomorphism l(Φ) : (O^{V} 𝔸^{n}, π^{−1}(0)) → (O^{V} 𝔸^{n}, π^{−1}(0)) given by the relations

\{x^{i}, X^{∘}^{i}, X^{∗}^{i}\} ↦ \{y^{i} = ϕ̄^{i}(x^{j}), Y^{∘} ^{i} = X^{∘}^{i}, Y^{∗} ^{i} = ϕ^{∘}^{i}(x^{j}) + ∑_{|p|=1}^{q} \frac{1}{p!} \frac{D^{p}ϕ^{i}}{Dx^{p}} X^{∗}^{p}\}. (7)

Let H be the chart corresponding to the fixed chart h which was used in the definition of holonomy representation, and Diff_{𝔸,pr}(O^{V} M^{𝔸}, π^{−1}(X_{0})) the group of germs of projectable bundle diffeomorphisms which have the form (7) with respect to charts of the atlas {H_{α}, π^{−1}(U_{α})}. We define the holonomy representation

α_{O} : π_{1}(L, X_{0}) → Diff_{𝔸,pr}(O^{V} M^{𝔸}, π^{−1}(X_{0}))

as follows: α_{O}([γ]) = H^{−1}l(α(γ))H. Passing to the principle bundle P^{V} M^{𝔸} we define in the same way the holonomy representation

α_{P} : π_{1}(L, X_{0}) → Diff_{𝔸,pr}(P^{V} M^{𝔸}, π^{−1}(X_{0})).

§3. 𝔸-affine horizontal distributions on M^{𝔸}

The bundle J^{𝔸}M can be treated as a bundle with structure group D_{n}(𝔸) [8]. A connection ∇ in the corresponding principal D_{n}(𝔸)-bundle is called an 𝔸-affine connection. The horizontal distribution on J^{𝔸}M induced by the connection ∇ is given in local 𝔸-coordinates {X^{i}} on J^{𝔸}M by the equations

dX^{∘}^{i} − ∑_{|p|=0}^{q} Γ_{j p}^{i}X^{∘}^{p} dx^{j} = 0  (8)

where Γ_{j p}^{i} = Γ_{j p}^{i}(x^{k}) are 𝔸/(𝔸^{∘}^{n})^{q−|p|+1}-valued connection coefficients [8]. The transformation law of the coefficients Γ_{j p}^{i} is derived in [14].

Definition[14]. A distribution ∇ on an 𝔸-smooth manifold M^{𝔸} complementary

to leaves of the canonical 𝔸^{∘}-foliation is called an 𝔸-affine horizontal distribution if in terms of 𝔸-charts (U_{α}, h_{α}), α ∈ A, it is given by equations of the form (8),

where Γ_{j p}^{i} are projectable functions with respect to the canonical 𝔸^{∘}-foliation.

An 𝔸-affine horizontal distribution (8) determines a horizontal distribution on the bundle O^{V} M^{𝔸} given by the equations

dX^{∗}^{i} − ∑_{|p|=0}^{q} Γ_{j p}^{i}X^{∗}^{p} dx^{j} = 0  (9)

in terms of charts (π^{−1}(U_{α}), H_{α}), α ∈ A. The distribution (9) is associated with

a projectable [5] (with respect to the canonical 𝔸^{∘}-foliation) connection ∇^{V} in the foliated principal bundle P^{V} M^{𝔸}. Conversely, a projectable connection ∇^{V} in the principal bundle P^{V} M^{𝔸} determines uniquely an 𝔸-affine horizontal distribution, consequently M^{𝔸} admits an 𝔸-affine horizontal distribution if and only if there is a projectable connection in P^{V} M^{𝔸}. A necessary and sufficient condition for existence of such a connection is vanishing of the Atiyah class [5] a(P^{V} M^{𝔸}) in the d_{F} -cohomology group H_{ad F}^{1,1} (P^{V} M^{𝔸}, 𝔡_{n}(𝔸)) of tensorial forms of ad-type on P^{V} M^{𝔸} where ad is the adjoint representation of the group D_{n}(𝔸) in its Lie algebra 𝔡_{n}(𝔸).

The difference ω_{1} − ω_{2} of connection forms of two projectable connections ∇_{1}^{V} and ∇_{2}^{V} is a tensorial form of type (0, 1), which can be treated as a d_{F} -cohomology class belonging to H_{ad F}^{0,1} (P^{V} M^{𝔸}, 𝔡_{n}(𝔸)). Therefore, in case when M^{𝔸} admits an 𝔸-affine horizontal distribution, the set of all such distributions is in one-to-one correspondence with the group H_{ad F}^{0,1} (P^{V} M^{𝔸}, 𝔡_{n}(𝔸)).

The holonomy of M^{𝔸} may be an obstruction to existence of an 𝔸-affine horizontal distribution. Let Z_{0} be an element of a fiber P_{X_{0}}^{V}M^{𝔸} over X_{0} ∈ M^{𝔸}, Q_{Z_{0}} ⊂ T_{Z_{0}}P^{V} M^{𝔸} the horizontal subspace of projectable connection ∇^{V}, and [γ] ∈ π_{1}(L, X_{0}). Then the tangent mapping T_{Z_{0}}α_{P} (γ) maps Q_{Z_{0}} into Q_{R_{g}(Z_{0})} where g ∈ D_{n}(𝔸) is the element determined by the relation α_{P} (γ)(Z_{0}) = R_{g}(Z_{0}), hence Q_{Z_{0}} is invariant under the mapping T_{Z_{0}}α_{P} (γ) ∘ R_{g^{−1}}. All such mappings constitute a subgroup α_{P,Z_{0}} ⊂ GL(T_{Z_{0}}P^{V} M^{𝔸}).

Summing up the above discussion we obtain the following

Theorem 2.

(1) [14] In order that on M^{𝔸} there exist an 𝔸-affine horizontal distribution it is necessary and sufficient that the Atiyah-Molino class

a(P^{V} M^{𝔸}) ∈ H_{ad F}^{1,1} (P^{V} M^{𝔸}, 𝔡_{n}(𝔸))

vanish.

(2) If a(P^{V} M^{𝔸}) = 0, 𝔸-affine horizontal distributions are in one-to-one correspondence with elements of H_{ad F}^{0,1} (P^{V} M^{𝔸}, 𝔡_{n}(𝔸)).

(3) In order that on M^{𝔸} there exist an 𝔸-affine horizontal distribution it is necessary that for each Z_{0} ∈ P^{V} M^{𝔸} there exist a subspace Q_{Z_{0}} ⊂ T_{Z_{0}}P^{V} M^{𝔸} which is invariant by α_{P,Z_{0}}.

(4) For existence of an 𝔸-affine horizontal distribution it is necessary that there exist a projectable connection in B_{tr}^{q}M^{𝔸}, that is

a(B_{tr}^{q}M^{𝔸}) ∈ H_{ad F}^{1,1} (B_{tr}^{q}M^{𝔸}, 𝔤_{n}^{q})

vanish.

Examples. 1. An 𝔸-affine connection in the bundle J^{𝔸}M over a real manifold M is a trivial example of an 𝔸-affine horizontal distribution.

2. The quotient group of the additive group 𝔸^{n} by a discrete subgroup G ⊂ 𝔸^{n} generated by a real basis of 𝔸^{n} is the n-dimensional torus over 𝔸 [9]. Since the standard flat connection in 𝔸^{n} = J^{𝔸}ℝ^{n} is invariant by G, n-dimensional tori admit 𝔸-affine horizontal distributions.

Note. As it was shown in [14] in the case of complete manifold M^{𝔸} an 𝔸-affine horizontal distribution is an Ehresmann connection in the sense of R. A. Blumental-J. J. Hebda [15]. Foliations with Ehresmann connections were also studied by N. I. Zhukova in [16].

§4. 𝔸-smooth connections on M^{𝔸}

In this section the problem of existence of 𝔸-smooth connections in 𝔸-smooth principal bundle is considered. Firstly we describe the d̂-cohomology [17], which generalizes on the one hand the Dolbeault cohomology of complex manifolds and on the other hand the foliated cohomology developed by I.Vaisman [18] and P.Molino [5].

Let Λ^{p}M^{𝔸} be the bundle of real exterior p-forms on M^{𝔸}, 𝔸 ⊗ Λ^{p}M^{𝔸} the bundle of 𝔸-valued p-forms, Λ_{𝔸−lin}^{p}M^{𝔸} the subbundle of 𝔸 ⊗ Λ^{p}M^{𝔸} consisting of 𝔸-linear p-forms, and 𝔸⊗Λ^{1}M^{𝔸} = Λ_{𝔸−lin}^{1}M^{𝔸} ⊕Λ̃^{1}M^{𝔸} an arbitrary decomposition into the Whitney sum of 𝔸-module bundles (which, for instance, can be constructed with the use of a differentiable nondegenerate 𝔸-linear metric on M^{𝔸}). The bundles Λ^{p,q}M^{𝔸} = Λ̃^{p}M^{𝔸} ⊗ Λ_{𝔸−lin}^{q}M^{𝔸} can be treated as subbundles of the bundle 𝔸 ⊗ Λ^{p+q}M^{𝔸}. By Ω^{p,q}(U), where U ⊂ M^{𝔸} is an open subset, we denote the 𝔸-module of local sections of Λ^{p,q}M^{𝔸} over U. A form θ ∈ Ω^{p,q}(U) will be called a (p, q)-form. The exterior differential dθ of a (p, q)-form θ lies in Ω^{p−1,q+2}(U) ⊕ Ω^{p,q+1}(U) ⊕ Ω^{p+1,q}(U), and the differential d̂θ is defined as the component of dθ belonging to Ω^{p+1,q}(U). There holds d̂∘ d̂= 0, and H^{p,q}M^{𝔸} denote the homology groups of the complex

0 ⟶ Ω^{0,q}(M^{𝔸}) ⟶^{d̂} Ω^{1,q}(M^{𝔸}) ⟶^{d̂} Ω^{2,q}(M^{𝔸}) ⟶^{d̂} ….  (10)

It follows from definitions that (0, q)-form θ is 𝔸-smooth if and only if d̂θ = 0.

The Molino’s construction [5] of d_{F} -cohomology of tensorial forms on a foliated principal bundle can be extended to the case of 𝔸-smooth principal bundle as follows. Let P^{𝔸}(M^{𝔸}, p, G^{𝔸}) be an 𝔸-smooth principal bundle over M^{𝔸} with 𝔸smooth structure Lie group G^{𝔸}, V^{𝔸} an 𝔸-module isomorphic to 𝔸^{k} for some k, GL_{𝔸}(V^{𝔸}) the group of 𝔸-linear transformations of V^{𝔸}, and ρ : G^{𝔸} → GL_{𝔸}(V^{𝔸}) an 𝔸-smooth representation of G^{𝔸} in V^{𝔸}. An r-form θ with values in V^{𝔸} is called tensorial [19] if R_{g}^{∗}θ = ρ(g^{−1})θ for g ∈ G^{𝔸}, and θ(v_{1}, …, v_{r}) = 0 if at least one of the vectors v_{1}, …, v_{r} is vertical. A tensorial (p+q)-form θ with values in V^{𝔸} is said to be a (p, q)-form if for every local section s : U ⊂ M^{𝔸} → P^{𝔸} s^{∗}θ is a (p, q)-form on M^{𝔸}. Let {s_{α} : U_{α} ⊂ M^{𝔸} → P^{𝔸}}_{α∈A} be a family of 𝔸-smooth sections where {U_{α}}_{α∈A} is a covering of M^{𝔸}. We have s_{β} = R_{g_{βα}} ∘ s_{α} where g_{βα} : U_{α} ∩ U_{β} → G^{𝔸} are 𝔸-smooth functions. Denote θ_{α} = s_{α}^{∗}θ. Then we have

θ_{β} = ρ(g_{βα}^{−1})θ_{α}.  (11)

Tensorial (p, q)-forms with values in V^{𝔸} on P^{𝔸} are in one-to-one correspondence with families {θ_{α}}_{α∈A} of local V^{𝔸}-valued forms on M^{𝔸} satisfying (11). Since ρ(g_{βα}^{−1}) is 𝔸-smooth, from (11) it follows that d̂θ_{β} = ρ(g_{βα}^{−1})d̂θ_{α}. Hence the family {d̂θ_{α}}_{α∈A} determines a tensorial (p+1, q)-form, which will be denoted by d̂θ. Thus we obtain the operator

d̂: Ω_{ρ}^{p,q}(P^{𝔸}, V^{𝔸}) → Ω_{ρ}^{p+1,q}(P^{𝔸}, V^{𝔸})

where Ω_{ρ}^{p,q}(P^{𝔸}, V^{𝔸}) is the 𝔸-module of tensorial (p, q)-forms with values in V^{𝔸}. The corresponding cohomology groups will be denoted by H_{ρ}^{p,q}(P^{𝔸}, V^{𝔸}).

Let now Γ be a connection in P^{𝔸} and

ω : T P^{𝔸} → 𝔤^{𝔸}  (12)

the connection form of Γ. If the mapping ω_{Z} : T_{Z}P^{𝔸} → 𝔤^{𝔸} is 𝔸-linear for each Z ∈ P^{𝔸}, then Γ is called an adapted connection, and if the mapping (12) is 𝔸-smooth, Γ is called an 𝔸-smooth connection. Repeating word for word the discussion from the section 2.6 of [5] we obtain that the (1, 1)-component Ω_{1,1} of the curvature form of an adapted connection is d̂-closed, the class a(P^{𝔸}) = [Ω_{1,1}] ∈ H_{ad}^{1,1}(P^{𝔸}, 𝔤^{𝔸}), where ad is the adjoint representation of G^{𝔸} in 𝔤^{𝔸}, does not depend on a choice of adapted connection, and there holds

Theorem 3. a(P^{𝔸}) = 0 if and only if there exists an 𝔸-smooth connection in P^{𝔸}.

The class a(P^{𝔸}) is called the Atiyah class of P^{𝔸}.

Our next aim is to establish relations between the d̂-cohomology and the d_{F} -

cohomology with respect to the canonical 𝔸^{∘}-foliation. Let T M^{𝔸} = T^{v}M^{𝔸} ⊕T^{h}M^{𝔸} be the decomposition of the tangent bundle of M^{𝔸} into the Whitney sum of

the submodule T^{v}M^{𝔸} formed by tangent vectors to leaves of the canonical 𝔸^{∘}- foliation and some complementary subbundle T^{h}M^{𝔸}. This decomposition induces the decomposition of the cotangent bundle Λ^{1}M^{𝔸} = Λ_{v}^{1}M^{𝔸} ⊕ Λ_{h}^{1}M^{𝔸}. If {e_{a}} =

{e_{0}, e_{1}, …, e_{m}} is a basis in 𝔸 such that e_{0} = 1, eâ ∈ 𝔸^{∘} where â = 1, …, m,

and X^{i} = x^{ia}e_{ia} = x^{i}e_{0} + x^{iâ}eâ are the expansions of local 𝔸-coordinates on M^{𝔸} in terms of the basis {e_{a}}, then T^{v}M^{𝔸} is spanned by ∂/∂x^{iâ}, T^{h}M^{𝔸} by ∂/∂x^{i} − μ_{i}^{jâ}∂/∂x^{jâ}, Λ_{v}^{1}M^{𝔸} by η^{iâ} = dx^{iâ} + μ_{j}^{iâ}dx^{j}, and Λ_{h}^{1}M^{𝔸} by dx^{i}. Let, in addition, the subbundle Λ̃^{1}M^{𝔸} is spanned by the forms θ^{iâ} = dx^{iâ} + λ_{j}^{iâ}dX^{j}. We define the following two mappings β_{1} : Λ_{𝔸−lin}^{1}M^{𝔸} ∋ ω ↦ ω̄ ∈ Λ_{h}^{1}M^{𝔸} and β_{2} : Λ̃^{1}M^{𝔸} ∋ ω̃ ↦ ω̂ ∈ Λ_{v}^{1}M^{𝔸} as follows: ω̄ = π_{0}^{q} ∘ ω and ω̂ = π_{0}^{q} ∘ ω̃ ∘ pr_{v} where pr_{v} : T M^{𝔸} → T^{v}M^{𝔸} is the projection. In local coordinates there hold f_{i}dX^{i} ↦ f_{i}^{0}dx^{i0} and f_{iâ}θ^{iâ} ↦ f_{iâ}^{0}η^{iâ} where f_{i} = f_{i}^{a}e_{a} and f_{iâ} = f_{iâ}^{b}e_{b} are the expansions in terms of the basis {e_{a}}. The mappings β_{1} and β_{2} induce the mappings β : Λ^{p,q}M^{𝔸} → Λ_{v}^{p}M^{𝔸} ⊗ Λ_{h}^{q}M^{𝔸} = Λ_{fol}^{p,q}M^{𝔸} and β : Ω^{p,q}M^{𝔸} → Ω_{fol}^{p,q}M^{𝔸} where Ω_{fol}^{p,q}M^{𝔸} is the vector space of smooth sections of the bundle Λ_{fol}^{p,q}M^{𝔸}. It can be easily verified that d_{F} ∘ β = β ∘ d̂, and so β is a morphism of the complex (10) into the foliated complex

0 ⟶ Ω_{fol}^{0,q}(M^{𝔸}) ⟶^{d_{F}} Ω_{fol}^{1,q}(M^{𝔸}) ⟶^{d_{F}} Ω_{fol}^{2,q}(M^{𝔸}) ⟶^{d_{F}} ….

Then we obtain the corresponding morphism of the cohomology groups

β^{∗} : H^{p,q}M^{𝔸} → H_{fol}^{p,q}M^{𝔸}.

This morphism β can be extended to the case of tensorial forms on principal

bundles in the following situation. Suppose the leaf G^{∘}_{e}^{𝔸} through the identity element e of the structure group G^{𝔸} of an 𝔸-smooth principal bundle P^{𝔸}(M^{𝔸}, p, G^{𝔸})

is a closed subgroup in G^{𝔸}, and let π : G^{𝔸} → G = G^{𝔸}/G^{∘}_{e}^{𝔸} be the canoni-

cal epimorphism. Then we can factorize P^{𝔸} by the action of G^{∘}_{e}^{𝔸} and obtain a quotient foliated principal bundle P_{tr}(M^{𝔸}, p_{tr}, G). Let π_{P} : P^{𝔸} → P denotes the canonical projection. An 𝔸-module V^{𝔸}, isomorphic to 𝔸^{k}, can be factor-

ized by the submodule corresponding to 𝔸^{∘}^{k} ⊂ 𝔸^{k}, let V be the corresponding real vector space, and π_{V} : V^{𝔸} → V the canonical ephimorphism (of vector spaces). In this case a representation ρ : G^{𝔸} → GL_{𝔸}(V^{𝔸}) induces a representation ρ_{tr} : G → GL_{ℝ}(V), and to each tensorial (p, q)-form θ determined by a family of local V^{𝔸}-valued (p, q)-forms {θ_{α}} we assign the form θ_{tr} determined by the family of foliated sections {π_{P} ∘ s_{α}}_{α∈A} and the family of local V -valued (p, q)-forms {θ_{tr,α} = π_{V} ∘ θ_{α}}. Thus we obtain the morphism of complexes β : Ω_{ρ}^{p,q}(P^{𝔸}, V^{𝔸}) → Ω_{ρ_{tr},F}^{p,q} (P_{tr}, V) and the corresponding morphism of cohomology groups β^{∗} : H_{ρ}^{p,q}(P^{𝔸}, V^{𝔸}) → H_{ρ_{tr},F}^{p,q} (P_{tr}, V).

Since, by definition, the connection form (12) of an adapted connection Γ in P^{𝔸} is 𝔸-linear on T_{Z}P^{𝔸}, there is a unique connection Γ_{tr} in P_{tr} with the connection form ω_{tr} : T P_{tr} → 𝔤 determined by the relation ω_{tr} ∘ T π_{P} = π_{𝔤} ∘ ω. Analyzing the structure equations of the connections ω and ω_{tr} (see [5], §2.6) and taking into account the structure of 𝔸-smooth mappings (see (2)) we obtain Theorem 4.

(1) If a connection Γ in P^{𝔸} is 𝔸-smooth, then the connection Γ_{tr} in P_{tr} is projectable.

(2) β^{∗}(α(P^{𝔸})) = α(P_{tr}).

Corollary.

(1) In order that there exists an 𝔸-smooth connection in P^{𝔸} it is necessary that α(P_{tr}) = 0.

(2) An 𝔸-smooth 𝔸-linear connection ∇^{𝔸} on an 𝔸-smooth manifold M^{𝔸} induces a transversal [5] connection ∇_{tr} on M^{𝔸} with respect to the canonical

𝔸^{∘}-foliation.

Notes. In the same way as in Section 2 taking compositions of transition functions

of an 𝔸-smooth bundle P^{𝔸} along loops lying in a leaf L of the canonical 𝔸^{∘}-foliation on M^{𝔸} through X_{0} ∈ L we obtain a homomorphism α of π_{1}(L, X_{0}) into the group of germs of automorphisms of P^{𝔸} over a neighborhood of X_{0}, and so in order that there exist an 𝔸-smooth connection in P^{𝔸} it is necessary that there exist a right-invariant horizontal distribution along L which is invariant under this representation. In the case when P^{𝔸} is a natural bundle [4] the homomorphism α is a holonomy representation, therefore the holonomy of M^{𝔸} may be an obstruction for existence of 𝔸-smooth connections in 𝔸-smooth natural bundles over M^{𝔸}.

Applying the Weil functor [4] J^{𝔸} to a real principal bundle P (M, p, G) we obtain, evidently, an 𝔸-smooth principal bundle, which admits 𝔸-smooth connections.

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Department of Mathematics, Kazan State University Lenina 18, 420008, Kazan, RUSSIA E-mail: shurygin@open.ksu.ras.ru