Lobachevskii Journal of Mathematics Vol. 9, 2001, 55 – 75 \begin{matrix}c \\ ◯ \end{matrix}Vadim V. Shurygin, Larisa B. Smolyakova

Vadim V. Shurygin, Larisa B. Smolyakova

AN ANALOG OF THE VAISMAN-MOLINO COHOMOLOGY

FOR MANIFOLDS MODELLED ON SOME TYPES OF MODULES OVER WEIL ALGEBRAS AND ITS APPLICATION

(submitted by B. N. Shapukov)

Abstract. An epimorphism μ : A → B of local Weil algebras induces the functor T^{μ} from the category of fibered manifolds to itself which assigns to a fibered manifold p : M → N the fibered product p^{μ} : T^{A}N ×_{T^{B}N} T^{B}M → T^{A}N. In this paper we show that the manifold T^{A}N ×_{T^{B}N} T^{B}M can be naturally endowed with a structure of an A-smooth manifold modelled on the A-module L = A^{n} ⊕ B^{m}, where n = dim N, n + m = dim M. We extend the functor T^{μ} to the category of foliated manifolds (M, F). Then we study A-smooth manifolds M^{L} whose foliated structure is locally equivalent to that of T^{A}N ×_{T^{B}N} T^{B}M. For such manifolds M^{L} we construct bigraduated cohomology groups which are similar to the bigraduated cohomology groups of foliated manifolds and generalize the bigraduated cohomology groups of A-smooth manifolds modelled on A-modules of the type A^{n}. As an application, we express the obstructions for existence of an Asmooth linear connection on M^{L} in terms of the introduced cohomology groups.

1. Introduction

Various jet bundles carry natural structures of smooth manifolds modelled on modules over local Weil algebras, that is, finite-dimensional algebras A

over R with unique maximal ideal A^{∘} such that the quotient algebra A/ A^{∘} is isomorphic to R. Examples of such jet bundles are: bundles of holonomic, nonholonomic and semiholonomic (N, q)-velocities of C. Ehresmann [5], [6], bundles of A-velocities (near points of A-type) of A.Weil [32], [18], [23], bundles of transverse A-velocities [33], [16], [19], semitangent bundles of V.V. Vishnevskii [29], frame bundles of higher order [4], [34]. In general, an A-smooth manifold M^{L} modelled on an A-module L possesses no fibered structure, but it

The research was supported in part by the Russian Foundation for Basic Research. Grant No 00-01-00308. carries canonical foliations corresponding to invariant submodules of the module L [20], [23]. The structure of an A-smooth manifold on M^{L} induces on leaves of the canonical foliations of M^{L} structures of (X, G)-manifolds in the sense of W.Thurston [26], and so M^{L} carries canonical structures of tangential (X, G)-foliations [9].

In this article, we study A-smooth manifolds modelled on A-modules of the type A^{n} ⊕ B^{m}, where B = A/I is the quotient algebra of A by an ideal I ⊂ A. A structure of an A-smooth manifold modelled on the module A^{n}⊕B^{m} arises naturally on the bundle T^{A}N ×_{T^{B}N} T^{B}M defined for a fibered manifold p : M → N, where n = dim N, n + m = dim M. In the case A = R(N, q), B = R(N, r), the corresponding bundle T^{A}N ×_{T^{B}N} T^{B}M belongs to the class of second order semitangent bundles studied by V.V. Vishnevskii [29]. The functor which assigns to a fibered manifold p : M → N the bundle T^{A}N ×_{T^{B}N} T^{B}M belongs to the class of so called product preserving functors. In more general situation of a not necessarily surjective homomorphisms μ : A → B such functors T^{μ} were studied by W.M. Mikulski [10] and J.V. Tomáš [27]. Similar functors T^{μ} can also be defined on the category of foliated manifolds (see §4 below). In the case of the canonical epimorphism μ : A → R, the functor T^{μ} on the category of foliated manifolds coincides with the transverse Weil functor T_{tr}^{A}. Thus, A-smooth manifolds modelled on A-modules of the type A^{n}⊕B^{m} are generalizations of manifolds with integrable almost transverse structures [3], [19].

Section 2 contains a brief description of basic notions of the theory of manifolds over local algebras.

In Section 3, we prove theorems on local structure of A-smooth mappings between A-modules in the following cases: 1) A^{n} → L, where A^{n} is the module of n-tuples of elements of A and L is an arbitrary A-module of finite dimension over R; 2) B^{n} → L, where B is the quotient algebra of A by an ideal I and L is an arbitrary A-module of finite dimension over R; 3) A^{n}⊕B^{m} → A^{n′} ⊕B^{m′}.

In Section 4, we introduce the notion of a μ-velocity on a foliated manifold (M, F), where μ : A → B is an epimorphism of local algebras and define the functor T^{μ}M which assigns to a foliated manifold (M, F) the bundle T^{μ}M of μ-velocities on (M, F) and to a morphism of foliations the naturally arising morphism of the bundles of μ-velocities. The bundle T^{μ}M can be described in terms of transverse Weil bundles, namely, there is a natural one-to-one correspondence T^{μ}M ↔ T_{tr}^{A}M ×_{T_{tr}^{B}M} T^{B}M. The bundle T^{μ}M of a foliated manifold (M, F) carries a natural structure of an A-smooth manifold modelled on the module A^{n}⊕B^{m}, where n+m = dim M and n is the codimension of the foliation F. In the case when the foliation F on M is determined by a surjective submersion p : M → N, T^{μ}M is naturally equivalent to T^{A}N ×_{T^{B}N} T^{B}M.

In this section we also define μ-lifts of projectable fields of foliated geometric objects from a foliated manifold (M, F) to its bundle T^{μ}M. As in the case of A-smooth manifolds modelled on A-modules of the type A^{n} and A^{n} ⊕ R^{m} [19], μ-lifts can be constructed using analytic prolongations of local equations of fields of geometric objects. As an example, we consider μ-lifts of projectable linear connections. Lifts of geometric objects to Weil bundles were studied by P.C. Yuen [35], L.-N. Patterson [15], A. Morimoto [13], A.Ya. Sultanov [25]. Lifts of geometric objects to transverse Weil bundles were studied by Z.Pogoda [16], R.Wolak [33], and in [19].

The pseudogroup of all local A-smooth diffeomorphisms between open subsets of the module A^{n} ⊕ B^{m} is too wide from geometrical point of view: it does not preserve canonical foliation A^{n} ⊕ B^{m} → A^{n}, and so an arbitrary A-smooth manifold modelled on A^{n} ⊕ B^{m} is not locally equivalent to the bundle T^{μ}R^{n+m}, where R^{n+m} is considered as a foliated manifold with foliation determined by the canonical projection R^{n+m} → R^{n}. By this reason we call an A-smooth mapping Φ : U ⊂ A^{n} ⊕ B^{m} → A^{n′} ⊕ B^{m′} foliated if it is foliated with respect to the foliations determined by the projections A^{n} ⊕ B^{m} → A^{n} and A^{n′} ⊕ B^{m′} → A^{n′}. An A-smooth manifold modelled on A^{n} ⊕ B^{m} is called foliated, if it is endowed with a maximal atlas whose coordinate transformations are foliated A-smooth diffeomorphisms. Then we generalise to the case of foliated A-smooth manifolds modelled on A-modules of the type A^{n} ⊕ B^{m} the bigraduated d̂-cohomology defined in [22] for A-smooth manifolds modelled on A-modules of the type A^{n}. Following [12], with the use of the d̂-cohomology of tensorial forms on the bundle of fibered A-linear frames on an A-smooth manifold M^{L} modelled on the A-module L = A^{n} ⊕ B^{m}, we construct the Atiyah-Molino class of M^{L} which is the obstruction for existence of a foliated A-smooth connection on M^{L}. The Atiyah classes as obstructions for existence of holomorphic complex connections in holomorphic principal bundles were constructed by M. Atiyah in [1]. Using d_{F} -cohomology of foliated manifolds [28], P. Molino constructed similar obstructions for existence of projectable connections in foliated principal bundles [11], [12]. I.V. Belko [2] generalized the Atiyah-Molino classes to the case of foliated Lie algebroids. In [24], the Atiyah-Molino classes which are obstructions for prolongation of transverse connections to A-smooth connections on an A-smooth manifold modelled on A^{n} were constructed.

2. Preliminaries

A finite-dimensional associative commutative unital algebra A over R is said to be local in the sense of A.Weil [32], [34], [8] if its radical (the set of all

nilpotent elements) Rad(A) =A^{∘} is a maximal ideal and the quotient algebra

A/A^{∘} is isomorphic to R. The one-dimensional vector subspace in A spanned by 1_{A} is a subalgebra isomorphic to R. We shall identify this subalgebra with R and suppose that R ⊂ A and 1_{A} ≡ 1 ∈ R. Then a local algebra A is represented as a semidirect sum

A = R ⊕ A^{∘} .  (1)

The structure of a local algebra A is completely determined by its radical, the

nilpotent algebra A^{∘}. A detailed exposition of the theory of local Weil algebras can be found in the articles of V.V.Wagner [30], [31] and in the book of I. Kolář, P.W. Michor, and J. Slovák [8]. One can choose a system of generators {ε^{A}},

A = 1, …, N, of the ideal A^{∘} such that the system of residue classes {ε^{A}+(A^{∘})^{2}} modulo the square (A^{∘})^{2} of A^{∘} is a basis of the quotient algebra A^{∘}/(A^{∘})^{2}. Any

such system {ε^{A}}, A = 1, …, N, is called a pseudobasis in A^{∘} [31], and the

number N = dim_{R}A^{∘}/(A^{∘})^{2} is called the width of A. The natural number q

determined by the relations (A^{∘})^{q} ≠ 0, (A^{∘})^{q+1} = 0 (here and in the sequel,

the symbol (A^{∘})^{r} denotes the r-th power of the ideal A^{∘}) is called the height of A. A local Weil algebra of width N and height q is isomorphic to a quotient algebra of the algebra R(N, q) of truncated polynomials of degree ≤ q in N variables (see, e. g., [23]).

We shall denote the structure constants of A with respect to a basis {e_{a}} by γ_{ab}^{c}.

Consider a composition series

A ⊃ A^{∘} = I_{1} ⊃ I_{2} ⊃ … ⊃ I_{m} ⊃ 0  (2)

which is a refinement of the series of ideals

A ⊃A^{∘}⊃ (A^{∘})^{2} ⊃ ⋯ ⊃ (A^{∘})^{r} ⊃ ⋯ ⊃ (A^{∘})^{q} ⊃ 0.

Since the quotient algebra I_{a}/I_{a+1} in (2) is one-dimensional and has zero multiplication, one can choose a basis

{e_{a}} = {e_{0}, eâ}, a = 0, 1, …, m = dim A^{∘}, â = 1, …, m,  (3)

in A such that e_{0} = 1 ∈ R, eâ ∈ Iâ, eâ ∉ I_{â+1}. The expansion of X ∈ A in terms of the basis (3) will be written as follows: X = x^{a}e_{a} = x^{0} + xâeâ. In what follows, we shall omit sometimes the superscript 0 and write x in place of x^{0}, x^{i} in place of x^{i0}, etc. The structure constants of A with respect to the basis (3) satisfy the following relations: γ_{0a}^{b} = δ_{a}^{b} (Kronecker’s deltas), γ_{ab̂}^{c} = 0 when a ≥ c.

Let L and L′ be two A-modules which are finite-dimensional real vector spaces. Let {E_{i}} and {E_{i′}} be bases in L and L′ as real vector spaces and

e_{a}E_{i} = τ_{ai}^{j}E_{j}, e_{a}E_{i′} = τ_{ai′}^{j′}E_{j′}  (4)

(we adopt Einstein’s summation convention). A smooth mapping Φ : U ⊂ L → L′ is said to be A-smooth if the tangent mappings

T_{X} : T_{X}U ≡ L → L′ ≡ T_{Φ(X)}L′  (5)

are A-linear for all X ∈ U. In terms of the coordinates {x^{i}} in L and {x^{i′}} in L′ corresponding to the bases {E_{i}} and {E_{i′}} respectively, the condition (5) takes the form

\frac{∂Φ^{i′}}{∂x^{k}} τ_{aj}^{k} = τ_{ak′}^{i′} \frac{∂Φ^{k′}}{∂x^{j}} ,  (6)

where τ_{aj}^{k} and τ_{ak′}^{i′} are determined by (4). The conditions of A-smoothness (6) of a mapping between A-modules are called the Scheffers equations [17].

An A-smooth manifold modelled on L (an L-manifold) is a smooth manifold endowed with an atlas whose charts h_{α} take values in the A-module L and such that the “coordinate transformations” h_{α}∘h_{β}^{−1} are A-smooth. Such charts and atlaces are called, respectively, L-charts and L-atlases (see, e. g., [23] for details).

By the symbol π_{r}^{q} we denote the natural epimorphism π_{r}^{q} : A → A_{(r)} =

A/(A^{∘})^{r+1}. In particular, π_{0}^{q} : A → R = A/A^{∘} is the epimorphism which assigns to an element X ∈ A its real part x ∈ R.

By A^{n} we denote the A-module of n-tuples of elements of A: A^{n} = A × A ⋯×A. The algebra epimorphism π_{r}^{q} induces the epimorphism of A-modules π_{r}^{q} : A^{n} → A_{(r)}^{n}, which is denoted by the same symbol.

Let I ⊂ A be an ideal, B = A/I the quotient algebra, and p : A → B the natural epimorphism. For the sake of brevity, we use the same symbol p for the epimorphism of modules p : A^{n} → B^{n} induced by p : A → B. The foliation on A^{n} determined by the submersion p is called the canonical I^{n}-foli-ation. The leaves of this foliation are residue classes modulo I^{n}. In particular,

the epimorphism π_{0}^{q} : A^{n} → R^{n} determines the canonical A^{n}^{∘}-foliation on A^{n}.

Recall that a smooth mapping f : M → W from a foliated manifold (M, F) is called projectable or basic (see, for example, [12]) if the values of f are constant along the leaves of the foliation F.

Theorem 1 [23]. 1) Let U ⊂ A^{n} be an open set and φ : U → A^{k} a mapping

projectable with respect to the canonical A^{n}^{∘}-foliation. Then the formula

X^{i′} = φ^{i′} + ∑_{|p|=1}^{q} \frac{1}{p!} \frac{D^{p}φ^{i′}}{Dx^{p}} X^{p}^{∘},  (7)

where i = 1, …, n, i′ = 1, …, k, p = (p_{1} …, p_{n}) is a multiindex of length n,

x^{i} = x^{i0}, X^{p}^{∘}= (X^{∘}^{1})^{p_{1}} … (X^{∘}^{n})^{p_{n}}, determines an A-smooth mapping Φ : U → A^{k}.

2) Any A-smooth mapping Φ : U → A^{k} is of the form (7) for some basic functions φ^{i′} : U → A.

3) If U is a simple open set for the canonical A^{n}^{∘}-foliation, then an A-smooth mapping Φ : U → A^{k} can be uniquely extended to an A-smooth one

Φ̃ : (π_{0}^{q})^{−1}(π_{0}^{q}(U)) → A^{k},

and the mappings φ and Φ̃ are related as follows: φ = (Φ̃|π_{0}^{q}(U)) ∘ π_{0}^{q} .

From the above theorem it follows that the Weil bundle T^{A}M_{n} of A-ve-locities over a smooth manifold M_{n} [32], [8], [23] is an A-smooth manifold modelled on the A-module A^{n}. This fact was discovered by A.P.Shirokov in [18].

3. Smooth mappings between modules over local Weil algebras

In this section, we prove theorems on the local structure of A-smooth mappings between A-modules of certain types which generalize Theorem 1. All modules over Weil algebras under consideration are supposed to be finite-dimensional real vector spaces.

Theorem 2. 1) Let U ⊂ A^{n} be an open set, L an A-module and φ : U → L

a mapping projectable with respect to the canonical A^{n}^{∘}-foliation of A^{n}. Then the formula

Y = φ + ∑_{|p|=1}^{q} \frac{1}{p!} X^{p}^{∘} \frac{D^{p}φ}{Dx^{p}},  (8)

where p = (p_{1} …, p_{n}) is a multiindex of length n, x^{i} = x^{i0}, X^{p}^{∘}= (X^{∘}^{1})^{p_{1}} … (X^{∘}^{n})^{p_{n}}, determines an A-smooth mapping Φ : U → L.

2) Any A-smooth mapping Φ : U → L is of the form (8) for some basic mapping φ : U → L.

3) If U is a simple open set for the canonical A^{n}^{∘}-foliation, then an A-smooth mapping Φ : U → L can be uniquely extended to an A-smooth mapping

Φ̃ : (π_{0}^{q})^{−1}(π_{0}^{q}(U)) → L,

and the mappings φ and Φ̃ are related as follows: φ = (Φ̃|π_{0}^{q}(U)) ∘ π_{0}^{q} .

Proof. 1) A mapping Φ : U ∋ {X^{i}} ↦ Y = Φ(X) ∈ L is an A-smooth if its differential dΦ is an L-valued A-linear 1-form on U, which is equivalent to the fact that dΦ = dX^{i}Φ_{i}. Let

Φ(X^{i}) = ∑_{|p|=0}^{q} \frac{1}{p!} X^{p}^{∘} \frac{D^{p}φ}{Dx^{p}},  (9)

where, for the sake of brevity, we put D^{0}φ/Dx^{0} = φ, and as usually, 0! = 1,

X^{∘}^{0} = 1. Similarly to the proof of Theorem 1 in [23], we have

dΦ = ∑_{i=1}^{n} ∑_{|p|=0}^{q} \frac{1}{p!} X^{p}^{∘} dx^{i} \frac{D^{p+i}φ}{Dx^{p+i}} + ∑_{i=1}^{n} ∑_{|p|=0}^{q} \frac{1}{p!}p_{i}X^{∘}^{p−i}dX^{∘}^{i} \frac{D^{p}φ}{Dx^{p}} =

∑_{i=1}^{n} dx^{i} (∑_{|p|=0}^{q} \frac{1}{p!} X^{p}^{∘} \frac{D^{p}}{Dx^{p}} (\frac{∂φ}{∂x^{i}} )) + ∑_{i=1}^{n} dX^{∘}^{i} ( ∑_{|w|=0}^{q−1} \frac{1}{w!}X^{∘}^{w} \frac{D^{w}}{Dx^{w}} (\frac{∂φ}{∂x^{i}} )) ,

where w = p − i, that is, w_{j} = p_{j} for j ≠ i, w_{i} = p_{i} − 1, and it is supposed

that p_{i}X^{∘}^{p−i} = 0 when p_{i} = 0. Thus,

dΦ = ∑_{i=1}^{n} dX^{i} (∑_{|p|=0}^{q} \frac{1}{p!} X^{p}^{∘} \frac{D^{p}}{Dx^{p}} (\frac{∂φ}{∂x^{i}} )) .  (10)

To prove the second statement, consider the series of submodules of L:

L = L_{0} ⊃ I_{1} · L = L_{1} ⊃ I_{2} · L = L_{2} ⊃ … ⊃ I_{m} · L = L_{m} ⊃ 0

and choose a basis {E_{α}} = {E_{α_{a}}} such that E_{α_{a}} ∈ L_{a}, {E_{α_{a}} + L_{a+1}} is a basis in L_{a}/L_{a+1}, where the index a takes the values from 0 to m such that L_{a}/L_{a+1} ≠ 0. For the basis {E_{α}} chosen above, we have τ_{âα}^{β} = 0 when α ≥ β. The standard A-basis {E_{1} = (1; 0; …; 0), …, E_{n} = (0; 0; …; 1)} in A^{n} and the basis {e_{a}} in A induce the basis {E_{ia} = e_{a}E_{i}} in A^{n} as a real vector space. In terms of the basis {E_{ia}}, an element X ∈ A^{n} has an expansion X = X^{i}E_{i} = x^{ia}E_{ia}, and the constants τ_{c jb}^{ia} are of the form τ_{c jb}^{ia} = δ_{j}^{i}γ_{cb}^{a}. The conditions of A-smoothness (6) of a mapping between two A-modules take in the case under consideration the following form:

\frac{∂Φ^{α}}{∂x^{kc}} γ_{ab}^{c} = τ_{aβ}^{α} \frac{∂Φ^{β}}{∂x^{kb}} ,  (11)

Putting b = 0 in (11), we obtain the system of equations

\frac{∂Φ^{α}}{∂x^{ka}} = τ_{aβ}^{α} \frac{∂Φ^{β}}{∂x^{k0}} .  (12)

Show that the two systems of equations (11) and (12) are equivalent. The relation e_{a}(e_{b}E_{α}) = (e_{a}e_{b})E_{α} implies that

τ_{aβ}^{σ}τ_{bα}^{β} = γ_{ab}^{c}τ_{cα}^{σ}.  (13)

Making use of (12) and (13), we obtain

\frac{∂Φ^{α}}{∂x^{kc}} γ_{ab}^{c} = τ_{cσ}^{α} \frac{∂Φ^{σ}}{∂x^{k0}} γ_{ab}^{c} = τ_{aρ}^{α}τ_{bσ}^{ρ} \frac{∂Φ^{σ}}{∂x^{k0}} = τ_{aρ}^{α} \frac{∂Φ^{ρ}}{∂x^{kb}} .

Let us assume that the domain U is an open coordinate parallelepiped determined by the system of inequalities A^{ia} < x^{ia} < B^{ia}, where A^{ia} < 0 and B^{ia} > 0. Let us denote φ = Φ|R^{n} ∩ U (recall that we suppose that R ⊂ A, which allows to consider R^{n} as a vector subspace in A^{n}) and show that there exists a unique A-smooth mapping Ψ : U → L such that

Ψ|R^{n} ∩ U = φ.  (14)

The mapping Ψ(X^{i}) = Ψ^{β}(x^{ib})E_{β} satisfies the Scheffers equations (12)

∂_{ia}Ψ^{β} = τ_{aσ}^{β}∂_{i0}Ψ^{σ}.  (15)

For the sake of brevity, here and in what follows we write ∂_{ia}f for ∂f/∂_{ia}. For β = β_{0} in (15), we obtain the following relations ∂_{i0}Ψ^{β_{0}} = ∂_{i0}Ψ^{β_{0}} and ∂_{ia}Ψ^{β_{0}} = 0 when a > 0. Hence it follows that Ψ^{β_{0}} depends only on x^{i0}, therefore, Ψ^{β_{0}}(x^{i0}) = φ^{β_{0}}(x^{i0}). For the function Ψ^{β_{b}}(x^{ia}), where b > 0, from (15) we obtain the following system of equations:

∂_{ia}Ψ^{β_{b}} = ∑_{g=2}^{b−1} τ_{aσ_{g}}^{β_{b}}∂_{i0}Ψ^{σ_{g}} for b > a > 0,

∂_{ib}Ψ^{β_{b}} = ∑_{σ_{0}} τ_{bσ_{0}}^{β_{b}}∂_{i0}Ψ^{σ_{0}} for b = a > 0, ∂_{ia}Ψ^{β_{b}} = 0 for a > b.

In particular, we have the following system of equations for the function Ψ^{β_{1}}:

∂_{i1}Ψ^{β_{1}} = ∑_{σ_{0}} τ_{1σ_{0}}^{β_{1}}∂_{i0}Ψ^{σ_{0}}, ∂_{ia}Ψ^{β_{1}} = 0 for a > 1.

Then

Ψ^{β_{1}} = x^{i1}f_{i1}^{β_{1}}(x^{i0}) + g^{β_{1}}(x^{i0}),  (16)

where f_{i1}^{β_{1}}(x^{i0}) = ∑_{σ_{0}} τ_{1σ_{0}}^{β_{1}}∂_{i0}Ψ^{σ_{0}}. Putting in (16) x^{i1} = 0 and taking into account (14), we obtain g^{β_{1}}(x^{i0}) = φ^{β_{1}}(x^{i0}). Consequently,

Ψ^{β_{1}} = x^{i1}f_{i1}^{β_{1}}(x^{i0}) + g^{β_{1}}(x^{i0}).  (17)

Taking the formula (17) as the basis for mathematical induction, let us show that there exists a unique function Ψ^{β_{b}} (b = 2, …, m) satisfying (14) and the Scheffers equations, and that Ψ^{β_{b}} is a polynomial in x^{ia}, a ≤ b, with coefficients depending on x^{i0}. Assuming that this statement holds for all β_{a} such that a < b, we obtain the following system of equations for ψ^{β_{b}}:

∂_{ia}ψ^{β_{b}} = P_{ia}^{β_{b}}, a ≤ b,  (18)

where P_{ia}^{β_{b}} are polynomials in x^{jg}, 0 < g < b, with coefficients depending on x^{j0}. It can be verified by direct calculations that the integrability conditions of the system (18) are fulfilled: ∂_{jc}P_{ia}^{β_{b}} = ∂_{ia}P_{jc}^{β_{b}}. But this fact also follows from the fact that the mapping Φ^{β}(x^{ia})E_{β} given by the equations (9) satisfies the equations (15). The solution of the system (18) is a polynomial in x^{ia}, 0 < a ≤ b, and is determined up to a summand depending on x^{i0}. The requirement Ψ^{β_{b}}|R^{n} ∩ U = φ^{β_{b}} determines the function Ψ^{β_{b}} uniquely. Thus, there exists only one A-smooth mapping Ψ : U → L satisfying the relation (14). Hence it follows that Ψ coincides with the mapping (9).

The remainder of the proof is similar to that of Theorem 1 in [23]. □

From (10) it follows that the differential dΦ of an A-smooth mapping Φ : U → L, viewed as an L-valued 1-form on U, is expanded in terms of the forms dX^{i}: dΦ = dX^{i}Φ_{i}, where

Φ_{i} = ∑_{|p|=0}^{q} \frac{1}{p!} X^{p}^{∘} \frac{D^{p}}{Dx^{p}} (\frac{∂φ}{∂x^{i}} ) .  (19)

By Theorem 2, the mapping Φ_{i} : U → L defined by (19) is A-smooth. We shall denote it as follows: Φ_{i} = ∂Φ/∂X^{i} and call the partial derivative of Φ with respect to the variable X^{i} ∈ A. It is easily seen from (19) and (9) that ∂Φ/∂X^{i} coincides with the partial derivative ∂Φ/∂x^{i} of Φ viewed as an L-valued function on U.

Let I ⊂ A be an ideal of a local Weil algebra A and B = A/I the quotient algebra. The B-module B^{n} of n-tuples of elements of B can also be regarded as an A-module where αX = 0 for any α ∈ I and X ∈ B^{n}. For an A-module L and an ideal I of A, by Ann_{L}(I) we shall denote the submodule in L consisting of the elements which are annihilated by all elements of I. The multiplication A × L → L induces a multiplication B × Ann_{L}(I) → Ann_{L}(I) so that Ann_{L}(I) can be regarded as a B-module.

Theorem 3. 1) Let U ⊂ B^{n} be an open set, L an A-module and φ : U → L

a mapping projectable with respect to the canonical B^{n}^{∘}-foliation of B^{n} and such that d_{X}φ takes values in Ann_{L}(I) for all X ∈ U. Then the formula

Y = φ + ∑_{|p|=1}^{q} \frac{1}{p!} X^{p}^{∘} \frac{D^{p}φ}{Dx^{p}}  (20)

determines an A-smooth mapping Φ : U → L.

2) Any A-smooth mapping Φ : U → L is of the form (20) for some basic mapping φ : U → L such that d_{X}φ takes values in Ann_{L}(I) for all X ∈ U.

3) If U is a simple open set for the canonical B^{n}^{∘}-foliation, then an A-smooth mapping Φ : U → L can be uniquely extended to an A-smooth mapping

Φ̃ : (π_{0}^{r})^{−1}(π_{0}^{r}(U)) → L,

where r is the height of B, and the mappings φ and Φ̃ are related as follows: φ = (Φ̃|π_{0}^{r}(U)) ∘ π_{0}^{r} .

Proof. Let ψ : B^{n} → L be an A-linear mapping. For arbitrary α ∈ I and X ∈ B^{n}, we have 0 = ψ(αX) = αψ(X). Thus, ψ takes values in Ann_{L}(I). Since a mapping ψ : B^{n} → Ann_{L}(I) is A-linear if and only if it is B-linear, the statements of Theorem 3 are consequences of the corresponding statements of Theorem 2. □

The next theorem establishes the local structure of A-smooth mappings between A-modules of the form A^{n}⊕B^{m}, where B = A/I is the quotient algebra by an ideal I. By μ : A → B we shall denote the canonical epimorphism. The quotient algebra B and the B-module B^{m} can be regarded as A-modules, therefore, the vector space L = A^{n} ⊕ B^{m} carries a natural structure of an A-

module. By L^{∘} we shall denote the submodule A^{n}^{∘} ⊕ B^{m}^{∘}⊂ L, by pr_{1} : L → A^{n} and pr_{2} : L → B^{m} the canonical projections, and by Ann (I) the annihilator Ann_{A}(I) of the ideal I in A.

Let us choose a basis

{e_{a}} = {e_{0} = 1, e_{a^{∗}}, eǎ}  (21)

in A such that {e_{a^{∗}}, eǎ} is a basis of the ideal A^{∘} and {eǎ} a basis of the ideal I. Then the collection \{e\overline{a}\} = \{e_{0} = 1, e_{a^{∗}}\} determines a basis in the quotient algebra B, which for the sake of simplicity we shall denote by the same symbols. An arbitrary element X ⊕ Y of the module A^{n} ⊕ B^{m} is determined by its

coordinates X^{i} = x^{i}+ X^{i}^{∘}∈ A, Y^{α} = y^{α}+ Y^{α}^{∘}∈ B, where x^{i}, y^{α} ∈ R, X^{i}^{∘}∈ A^{∘},

Y^{α}^{∘}∈ B^{∘}, and the indices run through the following ranges i, j, ⋯ = 1, 2, …, n, α, β, ⋯ = 1, 2, …, m.

Theorem 4. Let W = W_{1} ⊕ W_{2} ⊂ A^{n} ⊕ B^{m} be an open coordinate parallelepiped with respect to the real coordinates in A^{n} ⊕ B^{m} determined by the basis (21) in A. Then a mapping Φ : W → A^{n′} ⊕ B^{m′} is A-smooth if and only if it is of the following form:

X^{i′} = φ^{i′}(x^{i}) + ∑_{|p|=1}^{q} \frac{1}{p!} \frac{D^{p}φ^{i′}}{Dx^{p}} X^{p}^{∘} +ψ^{i′}(x^{i}, y^{α}) + ∑_{|u|+|v|=1}^{q} \frac{1}{u!v!} \frac{D^{u+v}ψ^{i′}}{Dx^{u}Dy^{v}} X^{u}^{∘}Y^{v}^{∘}, (22)

Y^{α′} = φ^{α′}(x^{i}, y^{α}) + ∑_{|u|+|v|=1}^{q} \frac{1}{u!v!} \frac{D^{u+v}φ^{α′}}{Dx^{u}Dy^{v}} X^{u}^{∘}Y^{v}^{∘},  (23)

where u, v, and p are multiindices, q the height of A, and the functions φ^{i′}(x^{i}), φ^{α′}(x^{i}, y^{α}), and ψ^{i′}(x^{i}, y^{α}) take values, respectively, in A, B, and Ann (I).

Proof. A mapping ξ : A^{n} ⊕ B^{m} → A^{n′} ⊕ B^{m′} is A-linear if and only if it is the sum of A-linear mappings ξ_{1} : A^{n} → A^{n′}, ξ_{2} : B^{m} → A^{n′}, ξ_{3} : A^{n} → B^{m′}, and ξ_{4} : B^{m} → B^{m′} such that ξ(X ⊕ Y) = (ξ_{1}(X) + ξ_{2}(Y)) ⊕ (ξ_{3}(X) + ξ_{4}(Y)). Consequently, a smooth mapping Φ : W → A^{n′} ⊕ B^{m′} is A-smooth if and only if both the mappings Φ_{1} = pr_{1} ∘ Φ : W → A^{n′} and Φ_{2} = pr_{2} ∘ Φ : W → B^{m′} are A-smooth. The mapping Φ_{1} is A-smooth if and only if all the mappings Φ_{1}|_{W_{1}⊕Y} and Φ_{1}|_{X⊕W_{2}} are A-smooth. Since Φ_{1}|_{W_{1}⊕Y} is A-smooth for any Y ∈ W_{2}, from Theorem 1 it follows that Φ_{1} is of the form:

X^{i′} = σ^{i′}(x^{i}, Y^{α}) + ∑_{|p|=1}^{q} \frac{1}{p!} \frac{D^{p}σ^{i′}(x^{i}, Y^{α})}{Dx^{p}} X^{p}^{∘} .  (24)

For any fixed X ∈ W_{1}, the formula (24) gives an A-smooth mapping Φ_{1}|_{X⊕W_{2}}, then from Theorem 3 it follows that in (24) the functions ν(σ^{i′}), where ν : A → A/Ann (I), do not depend on Y^{α}, that is, σ^{i′}(x^{i}, Y^{α}) = φ^{i′}(x^{i}) + ψ^{i′}(x^{i}, Y^{α}), where the functions ψ^{i′}(x^{i}, Y^{α}) take values in Ann (I), and the mapping Φ_{1} is of the form

X^{i′} = φ^{i′}(x^{i}) + ∑_{|p|=1}^{q} \frac{1}{p!} \frac{D^{p}φ^{i′}}{Dx^{p}} X^{p}^{∘} +ψ^{i′}(x^{i}, Y^{α}) + ∑_{|p|=1}^{q} \frac{1}{p!} \frac{D^{p}ψ^{i′}(x^{i}, Y^{α})}{Dx^{p}} X^{p}^{∘} . (25)

Assume first that the set W contains 0 ∈ L. In this case, putting in (25) X^{i}^{∘}= 0, we conclude that for any x^{i} = x_{0}^{i} the functions ψ^{i′}(x^{i}, Y^{α}) are A-smooth and so these functions are of the form

ψ^{i′}(x^{i}, Y^{α}) = ψ^{i′}(x^{i}, y^{α}) + ∑_{|p|=1}^{q} \frac{1}{p!} \frac{D^{p}ψ^{i′}(x^{i}, y^{α})}{Dy^{p}} Y^{p}^{∘} .  (26)

Substituting (26) into (25), we obtain the equations (22). If W does not contain 0 ∈ L, it suffices to take any X_{0} ⊕Y_{0} ∈ W and consider the composition Φ∘τ : W′ → A^{n′}⊕B^{m′} of Φ with the translation τ : W′ ∋ X⊕Y ↦ (X+X_{0})⊕(Y +Y_{0}) (cf. the proof of Theorem 1 in [23]).

Using similar arguments, we derive the formula (23) from Theorem 2. □

4. A-smooth manifolds modelled on modules of the type A^{n} ⊕ B^{m}

A foliated manifold (M, F) can be viewed as a manifold endowed with a foliated pseudogroup structure [12] which is defined as follows. The projection

p : R^{n+m} = R^{n} × R^{m} ∋ {x^{i}, y^{α}} ↦ {x^{i}} ∈ R^{n},

where the indices run through the same ranges as above, determines the standard foliation F^{(n)} of codimension n on the space R^{n+m} representing R^{n+m} as a disjoint union of m-dimensional submanifolds (leaves of the standard foliation)

R^{n+m} = ⋃_{x∈R^{n}}{x} × R^{m}.

A local diffeomorphism

φ : U ∋ {x^{i}, y^{α}} ↦ {φ^{j}(x^{i}, y^{α}), φ^{β}(x^{i}, y^{α})} ∈ U′,  (27)

where U and U′ are open subsets in R^{n+m}, is called a local automorphism of the standard foliation of codimension n on R^{n+m} if

∂φ^{j}/∂y^{α} = 0.  (28)

The set of all local automorphisms of the standard foliation is a pseudogroup, denote it by Γ_{n,m}. A foliation F of codimension n on an (n + m)-dimensional manifold M is determined by a maximal atlas Φ_{n,m} on M with coordinate transformations belonging to the pseudogroup Γ_{n,m} [12]. The charts of the atlas Φ_{n,m} are called foliated charts. A foliation F on M induces a foliation F_{U} on every open subset U ⊂ M. Connected open subset U ⊂ M is said to be simple for a foliation F if the foliation F_{U} on U is generated by a submersion p_{U} : U → \overline{U} and each inverse image p_{U}^{−1}(\overline{x}) is connected. A foliated chart (U, h) on (M, F) is called simple if U is a simple open subset in M. The leaf of a foliation F on M through a point x is the maximal connected submanifold L_{x} ∋ x in M (weakly embedded [12]) which is given in terms of simple foliated charts by equations of the form x^{i} = x_{0}^{i} = const.

Let (M, F) and (M′, F′) be two foliated manifolds and φ : M → M′ a smooth mapping. In terms of foliated charts on M and M′, φ is given by equations

x^{i′} = φ^{i′}(x^{i}, y^{α}), y^{α′} = φ^{α′}(x^{i}, y^{α}).  (29)

A mapping φ : M → M′ is a morphism of foliations if the functions φ^{i′} in (29) satisfy the condition ∂φ^{i′}/∂y^{α} = 0. A morphism of foliations maps leaves of the foliation F into leaves of the foliation F′.

Consider now a foliated manifold (M, F), dimM = n + m, codim F = n, and an epimorphism μ : A → B of local Weil algebras. Let T^{A}M and T^{B}M be the bundles, respectively, of A-velocities and of B-velocities on M [8], [20] (note that in [20] A-velocities were called A-jets, and in [14] the term A-jet was used in quite different sense), and let T_{tr}^{A}M be the bundle of transverse A-velocities on M [20], [16].

Definition. Two A-velocities j_{x}^{A}f and j_{x}^{A}g at x ∈ M are called μ-equi-valent if they determine the same transverse A-velocity j_{tr x}^{A}f = j_{tr x}^{A}g and the same B-velocity j_{x}^{B}f = j_{x}^{B}g at x ∈ M. The μ-equivalence class j_{x}^{μ}f of an A-velocity j_{x}^{A}f is called a μ-velocity at x ∈ M.

Denote the set of all μ-velocities on M by T^{μ}M. There is a natural one-to-one correspondence T^{μ}M ↔ T_{tr}^{A}M ×_{T_{tr}^{B}M} T^{B}M, and so T^{μ}M carries a natural structure of a smooth manifold. The natural projection T^{μ}M → M turns T^{μ}M into a locally trivial fiber bundle over M. We will call T^{μ}M the Weil bundle of μ-velocities on M.

If the foliation F on M is generated by a surjective submersion M → N (in this case M is said to be a fibered manifold [8]), the bundle T^{μ}M is naturally equivalent to the fibered product T^{A}N ×_{T^{B}N} T^{B}M.

Let (M, F) and (M′, F′) be two foliated manifolds. A morphism of foliations φ : M → M′ induces the bundle morphism T^{μ}φ : T^{μ}M ∋ j^{μ}f ↦ j^{μ}(φ ∘ f) ∈ T^{μ}M′. Assigning to a foliated manifold (M, F) the bundle T^{μ}M and to a morphism of foliations φ : M → M′ the bundle morphism T^{μ}φ, we obtain a functor from the category of foliated manifolds to the category of fiber bundles.

Proposition 5. The bundle T^{μ}M carries a natural structure of an Asmooth manifold modelled on the A-module A^{n} ⊕ B^{m}.

Proof. Applying the functor T^{μ} to a foliated chart h : U → R^{n+m} on M, we obtain the chart T^{μ}h : T^{μ}U → T^{μ}R^{n+m}. The bundle T^{μ}R^{n+m} = T^{A}R^{n}×_{T^{B}R^{n}} T^{B}R^{n+m} = A^{n} ×_{B^{n}} B^{n+m} can be naturally identified with A^{n} ⊕ B^{m}. A coordinate transformation h′ ∘ h ∈ Γ_{n,m} on M is of the form (27), where ∂φ^{j}/∂y^{α} = 0. Then the coordinate transformation T^{μ}h′ ∘ T^{μ}h on T^{μ}M is of the form (22), (23), where ψ^{i′}(x^{i}, y^{α}) = 0. □

In terms of foliated charts, a morphism of foliations φ : M → M′ is given by equations of the form (29), where ∂φ^{i′}/∂y^{α} = 0. In terms of the corresponding charts on the bundles T^{μ}M and T^{μ}M′, the mapping T^{μ}φ is given by the equations

X^{i′} = φ^{i′} + ∑_{|p|=1}^{q} \frac{1}{p!} \frac{D^{p}φ^{i′}}{Dx^{p}} X^{p}^{∘},  (30)

Y^{α′} = φ^{α′} + ∑_{|u|+|v|=1}^{q} \frac{1}{u!v!} \frac{D^{u+v}φ^{α′}}{Dx^{u}Dy^{v}} X^{u}^{∘}Y^{v}^{∘},  (31)

where the functions φ^{i′} and φ^{α′} take values in R, and so φ^{μ} is A-smooth. Thus, the functor T^{μ} can be viewed as a functor from the category of foliated manifolds to the category of A-smooth manifolds.

For any two foliated manifolds (M, F) and (M′, F′), the morphism φ^{μ} : T^{μ}M → T^{μ}M′ corresponding to a morphism of foliations φ : M → M′ commutes with the projections onto the bundles of transverse A-velosities, i. e., the following diagram commutes

T^{μ}M ↔ T_{tr}^{A}M ×_{T_{tr}^{B}M} T^{B}M ⟶^{T^{μ}φ} T_{tr}^{A}M′ ×_{T_{tr}^{B}M′} T^{B}M′ ↔ T^{μ}M′
pr_{1} ↓  ↓ pr_{1}
T_{tr}^{A}M  ⟶^{T_{tr}^{A}}  T_{tr}^{A}M′

(32)

For the A-modules L = A^{n} ⊕ B^{m} and L′ = A^{n′} ⊕ B^{m′} viewed as the bundles T^{μ}R^{n+m} and T^{μ}R^{n′+m′} respectively and a linear fibered mapping φ : R^{n+m} → T^{μ}R^{n′+m′} such that p′∘φ = φ∘p, where p : R^{n+m} → R^{n} and p′ : R^{n′+m′} → R^{n′} are natural projections, the diagram (32) takes the form

A^{n} ⊕ B^{m} ⟶^{T^{μ}φ} A^{n′} ⊕ B^{m′}
pr_{1} ↓  ↓ pr_{1}
A^{n}  ⟶^{φ}  A^{n′}

(33)

Definition. An A-linear mapping ξ : A^{n} ⊕ B^{m} → A^{n′} ⊕ B^{m′} is called fibered if it projects into an A-linear mapping \overline{ξ} : A^{n} → A^{n′}, i. e., there exists a mapping \overline{ξ} such that the following diagram commutes:

A^{n} ⊕ B^{m} ⟶^{ξ} A^{n′} ⊕ B^{m′}
pr_{1} ↓  ↓ pr_{1}
A^{n}  ⟶^{\overline{ξ}}  A^{n′}

(34)

Definition. An A-smooth mapping Φ : U ⊂ A^{n}⊕B^{m} → A^{n′} ⊕B^{m′} is called foliated if, for each X ∈ U, the tangent mapping d_{X} : A^{n} ⊕ B^{m} → A^{n′} ⊕ B^{m′} is a fibered A-linear mapping.

From the above discussion it follows that an A-smooth mapping Φ : U ⊂ A^{n} ⊕ B^{m} → A^{n′} ⊕ B^{m′} is foliated if locally it is given by equations of the form (22), (23), where ψ^{i′} = 0, i. e., by equations of the form (30), (31), where the functions φ^{i′} and φ^{α′} take values in A and B respectively.

Definition. An L-atlas A = {(U, h)} on an A-smooth manifold M^{L} modelled on an A-module L = A^{n} ⊕B^{m} is called foliated if the transition functions h′ ∘ h^{−1} between charts of this atlas are foliated A-smooth mappings. An Asmooth manifold M^{L} endowed with a maximal foliated L-atlas is called foliated.

A foliated A-smooth manifold M^{L} carries the canonical foliation F^{μ} whose leaves, in terms of charts of its foliated L-atlas, are given by the equations X^{i} = const.

The canonical foliation F^{μ} of the bundle T^{μ}M of a foliated manifold (M, F) will be called (following the terminology adopted in the theory of transverse bundles [12]) the lifted foliation.

Let U ⊂ R^{n+m}, where R^{n+m} is considered as a foliated manifold (R^{n+m}, F^{(n)}), φ : U → A^{n′} ⊕ B^{m′} be a morphism of foliations, and π_{0} : A^{n} ⊕ B^{m} ∋ {X^{i}, Y^{α}} ↦ {x^{i}, y^{α}} ∈ R^{n+m} the canonical projection. From the above discussion it follows that there exists a unique A-smooth mapping Φ : (π_{0})^{−1}(U) → A^{n′} ⊕B^{m′} whose restriction to U ⊂ (π_{0})^{−1}(U) coincides with φ. If φ has equations x^{i′} = φ^{i′}(x^{i}, y^{α}) y^{α′} = φ^{α′}(x^{i}, y^{α}), where ∂x^{i′}/∂y^{α} = 0, then the mapping Φ is given by the equations (30) and (31). We shall denote this mapping Φ by φ^{μ} and call it the analytic prolongation of φ.

From the uniqueness of the representation of an A-smooth mapping from W ⊂ A^{n} ⊕ B^{m} to A^{n′} ⊕ B^{m′} in the form (22), (23), one can easily derive the following properties of analytic prolongations.

Proposition 6.

1^{∘}. (φ + ψ)^{μ} = φ^{μ} + ψ^{μ}, 2^{∘}. (φ · ψ)^{μ} = φ^{μ} · ψ^{μ},

3^{∘}. (φ^{μ} ∘ ψ)^{μ} = φ^{μ} ∘ ψ^{μ},

4^{∘}. (D^{s+t}φ/Dx^{s}Dy^{t})^{μ} = D^{s+t}φ^{μ}/DX^{s}DY^{t}. □

Identifying T^{μ}R^{n+m} ≡ A^{n} ⊕ B^{m}, for a foliated mapping φ : U ⊂ R^{n+m} → R^{n′+m′} we obtain φ^{μ} = T^{μ}.

Fibered A-linear mappings ξ : L → L′ between modules L = A^{n} ⊕ B^{m} and L′ = A^{n′} ⊕ B^{m′} can be written in matrix form as follows

( \begin{matrix}X′ \\ Y′ \end{matrix} ) = ( \begin{matrix}ξ_{1} & 0 \\ ξ_{3} & ξ_{4} \end{matrix} ) ( \begin{matrix}X \\ Y \end{matrix} )  (35)

where ξ_{1} is a matrix with entries belonging to A and ξ_{3}, ξ_{4} matrices with entries belonging to B. Hence the module Hom′_{A}(L, L′) of all fibered A-linear mappings between L and L′ is an A-module isomorphic to A^{nn′} ⊕ B^{m′(n+m)}.

The vector space Hom′_{R}(R^{n+m}, R^{n′+m′}) of all fibered linear mappings between R^{n+m} and R^{n′+m′} is fibered over the vector space Hom_{R}(R^{n}, R^{n′}). Therefore, we can apply the functor T^{μ} to the space Hom′_{R}(R^{n+m}, R^{n′+m′}). One can easily see that the A-module T^{μ}Hom′_{R}(R^{n+m}, R^{n′+m′}) is isomorphic to the A-module Hom′_{A}(L, L′).

The Lie group GL′_{A}(L) of all fibered A-linear invertible mappings φ : L → L is an open subset in A^{n^{2}} ⊕ B^{m(n+m)}, it is isomorphic to the Lie group T^{μ}G_{n,m}, where G_{n,m} denotes the Lie group of all fibered invertible linear mappings R^{n+m} → R^{n+m}. The Lie algebra gl′_{A}(L) of the Lie group GL′_{A}(L) is an A-module isomorphic to A^{n^{2}} ⊕ B^{m(n+m)}.

The tangent space T_{X}M^{L} of a foliated A-smooth manifold M^{L} at X ∈ M^{L} is an A-module isomorphic to L, therefore, one can consider fibered A-lin-ear mappings from L to T_{X}M^{L}. The set B_{A}M^{L} of all fibered isomorphisms between L and T_{X}M^{L} carries a natural structure of an A-smooth manifold modelled on the A-module A^{n+n^{2}} ⊕ B^{m+m(n+m)}. B_{A}M^{L} is a principal fiber bundle over M^{L} with structure Lie group GL′_{A}(L). We shall call B_{A}M^{L} the bundle of fibered A-linear frames on M^{L}.

Let G_{n}^{r} be the Lie group of r-jets of all local diffeomorphisms of R^{n} at 0, G_{n,m}^{r} the Lie group of r-jets of all local automorphisms of the standard foliation F^{(n)} of codimension n on R^{n+m} at 0, and p : G_{n,m}^{r} → G_{n}^{r} the canonical projection. The set B_{fol}^{r}M of r-jets of all germs of local isomorphisms of foliations f : (R^{n+m}, 0) → (M, x) carries a natural structure of a principal fiber bundle over M with structure Lie group G_{n,m}^{r}. We call it the bundle of foliated r-frames of (M, F) [19]. Denote by B_{tr}^{r}M the principal bundle of tranverse r-frames of (M, F) [12], [20]. In terms of a foliated chart h : U ∋ x ↦ {x^{i}, y^{α}} ∈ R^{n+m} on (M, F), a germ of local isomorphism of foliations f : (R^{n+m}, 0) → (M, x) is given by equations x^{i} = f^{i}(t^{j}), y^{α} = f^{α}(t^{j}, s^{β}). A foliated chart h : U → R^{n+m} on (M, F) induces the chart h_{tr}^{r} :∋ j_{tr x}^{r}f ↦ {x^{i}, y^{α}, x_{u}^{i} = ∂^{|u|}f^{i}/∂t^{u}} on B_{tr}^{r}M, where u = (u^{1}, …, u^{n}) is a multiindex running through the set of values determined by 1 ≤ |u| ≤ r. The bundle B_{tr}^{r}M is a foliated bundle [12] and carries the so called lifted foliation F_{tr}^{r} whose leaves, in terms of induced charts, are given by the equations x^{i} = const, x_{u}^{i} = const. Assigning to the r-jet j^{r}f of a germ of local isomorphism of foliations f : (R^{n+m}, 0) → (M, x) the transverse r-jet j_{tr}^{r}(f ∘ i), where i : (R^{n}, 0) → (R^{n+m}, 0) is the germ at 0 ∈ R^{n} of the embedding {x^{i}} ↦ {x^{i}, 0}, we obtain the canonical epimorphism of principal fiber bundles p : B_{fol}^{r}M → B_{tr}^{r}M. A foliated chart h : U → R^{n+m} on (M, F) induces the chart h^{r} :∋ j_{x}^{r}f ↦ {x^{i}, y^{α}, x_{u}^{i} = ∂^{|u|}f^{i}/∂t^{u}, y_{u v}^{α} = ∂^{|u|+|v|}f^{α}/∂t^{u}∂s^{v}} on B_{fol}^{r}M, where u = (u^{1}, …, u^{n}), v = (v^{1}, …, v^{m}) are multiindices running trough the sets of values determined by 1 ≤ |u| + |v| ≤ r. The bundle B_{fol}^{r}M carries the canonical foliation F_{fol}^{r} whose leaves are inverse images of leaves of the foliation F_{tr}^{r} by the epimorphism p : B_{fol}^{r}M → B_{tr}^{r}M, and so the mapping p : B_{fol}^{r}M → B_{tr}^{r}M is a morphism of foliations. In terms of induced charts, leaves of F_{fol}^{r} are given by the equations x^{i} = const, x_{u}^{i} = const.

The fiber bundle T^{μ}M can be viewed as associated to B_{fol}^{q}M.

Denote by G_{n}^{r}(A) the Lie group of r-jets of all germs of A-smooth diffeomorphisms (A^{n}, 0) → (A^{n}, 0) and by G_{fol}^{r}(L) the Lie group of r-jets of all germs of A-smooth foliated diffeomorphisms of L = A^{n} ⊕ B^{m} at 0 ∈ L. Let M^{L} be a foliated A-smooth manifold modelled on L. One can easily see that the set B_{fol}^{r}(A)M^{L} of r-jets of all germs of foliated A-smooth diffeomorphisms (L, 0) → M^{L} is an A-smooth manifold modelled on an A-module of the type A^{n′} ⊕ B^{m′}. B_{fol}^{r}(A)M^{L} carries a natural structure of a principal fiber bundle over M^{L} with structure Lie group G_{fol}^{r}(L). We call it the bundle of foliated A-smooth r-frames of M^{L}. The principal bundle B_{fol}^{1}(A)M^{L} is naturally isomorphic to B_{A}M^{L}.

Let there be given a locally trivial fiber bundle ρ : F → \overline{F} and right actions of Lie groups α : G_{n,m}^{r} × F → F and \overline{α} : G_{n}^{r} × \overline{F} → \overline{F} such that the following diagram is commutative:

G_{n,m}^{r} × F ⟶^{α} F
p × ρ↓ ↓ρ
G_{n}^{r} × \overline{F} ⟶^{\overline{α}} \overline{F}

By a field of foliated geometric object of the ρ-type on (M, F) we shall mean an equivariant mapping λ : B_{fol}^{r}M → F such that for some equivariant mapping (a field of tranverse geometric object) \overline{λ} : B_{tr}^{r}M → \overline{F} the following diagram is commutative:

\begin{matrix}B_{fol}^{r}M & ⟶^{λ} & F \\ p↓ & & ↓ρ \\ B_{tr}^{r}M & ⟶^{\overline{λ}} & \overline{F} \end{matrix}  (36)

A field of foliated geometric object of the ρ-type λ : B_{fol}^{r}M → F on (M, F) is said to be projectable if the diagram (36) consists of morphisms of foliations (the manifold \overline{F} is assumed to be endowed with the trivial foliation whose leaves are points).

Let λ : B_{fol}^{r}M → F be a field of projectable geometric object on (M, F). Applying the functor T^{μ} to the diagram (36), we obtain the diagram (cf. [19], p. 137)

B_{fol}^{r}(A)T^{μ}M ⟶^{λ^{μ}} T^{μ}F
T^{μ}p↓  ↓T^{μ}ρ
B_{tr}^{r}(A)T^{μ}M ⟶^{\overline{λ}^{A}} T^{A}\overline{F}

(37)

where B_{tr}^{r}(A)T^{μ}M is the principal bundle of transverse A-smooth r-frames on T^{μ}M with respect to the canonical foliation F^{μ}, and the following natural equivalences are used: T^{μ}(B_{fol}^{r}M) ≅ B_{fol}^{r}(A)T^{μ}M, T^{μ}(B_{tr}^{r}M) ≅ B_{tr}^{r}(A)T^{μ}M, T^{μ}\overline{F} ≅ T^{A}\overline{F} (\overline{F} is regarded as a foliated manifold with the trivial foliation). The diagram (37) determines an A-smooth field of geometric object of the T^{μ}(F)-type on the bundle T^{μ}M, the μ-lift of the field λ. The mapping \overline{λ}^{A} : B_{tr}^{r}(A)T^{μ}M → \overline{F}^{A} determines an A-smooth transverse field of geometric object of the T^{A}\overline{F} -type on T^{μ}M, the transverse A-lift of the field \overline{λ}.

As an example, consider a projectable linear connection ∇ on (M, F). Such a connection can be viewed as a projectable field of second order geometric object Γ : B_{fol}^{2}M → F. In terms of a foliated chart on (M, F) (along the corresponding natural section of the bundle B_{fol}^{2}M), the field Γ is given by equations Γ_{jk}^{i}(x^{l}, y^{σ}), Γ_{jk}^{α}(x^{l}, y^{σ}), Γ_{jβ}^{α}(x^{l}, y^{σ}), Γ_{βk}^{α}(x^{l}, y^{σ}), Γ_{βγ}^{α}(x^{l}, y^{σ}), where ∂Γ_{jk}^{i}/∂y^{σ} = 0. The field Γ is projected into the field \overline{Γ} : B_{tr}^{2}M → \overline{F} given by the equations Γ_{jk}^{i}(x^{l}, y^{σ}). The field \overline{Γ} determines a projectable transverse connection on (M, F) [12]. The μ-lift of Γ is the A-smooth A-linear connection on T^{μ}M given by the equations Γ̃_{jk}^{i}(X^{l}, Y^{σ}), Γ̃_{jk}^{α}(X^{l}, Y^{σ}), Γ̃_{jβ}^{α}(X^{l}, Y^{σ}), Γ̃_{βk}^{α}(X^{l}, Y^{σ}), Γ̃_{βγ}^{α}(X^{l}, Y^{σ}), where Γ̃_{jk}^{i}(X^{l}, Y^{σ}) is expressed in terms of Γ_{jk}^{i}(x^{l}, y^{σ}) by the formulas similar to (30):

Γ̃_{jk}^{i}(X^{l}, Y^{σ}) = Γ_{jk}^{i} + ∑_{|p|=1}^{q} \frac{1}{p!} \frac{D^{p}Γ_{jk}^{i}}{Dx^{p}} X^{p}^{∘},  (38)

and the other Γ̃’s are expressed in terms of the corresponding Γ’s by the formulas similar to (31). For example,

Γ̃_{jk}^{α} = Γ_{jk}^{α} + ∑_{|u|+|v|=1}^{q} \frac{1}{u!v!} \frac{D^{u+v}Γ_{jk}^{α}}{Dx^{u}Dy^{v}} X^{u}^{∘}Y^{v}^{∘} .

The equations (38) determine a transverse A-smooth connection on T^{μ}M with respect to the canonical foliation F^{μ}.

5. d̂-cohomology of a foliated manifold M^{L}

Consider two A-modules L = A^{n}⊕B^{m}, L′ = A^{n′} ⊕B^{m′}. By A_{i}, i = 1, …, n, and B_{α}, α = 1, …, m, we shall denote, respectively, the i-th direct summand in L = A^{n} ⊕ B^{m} isomorphic to the regular A-module A and the α-th direct summand in L = A^{n} ⊕ B^{m} isomorphic to the A-module B. Let {e_{ia}} and \{e_{α\overline{a}}\} be real bases in A_{i} and B_{α}, respectively, corresponding to the basis (21) of A. The total system \{e_{ia}, e_{α\overline{a}}\}, i = 1, …, n, α = 1, …, m, is a real basis in L. We shall also use a similar notation for the A-module L′. A linear mapping θ : L → L′ can be regarded as an L′-valued 1-form on L. Denote by Λ_{R}^{1}(L, L′) = Hom_{R}(L, L′) the vector space of all linear mappings from L to L′. Clearly, Λ_{R}^{1}(L, L′) is an A-module and

Λ_{R}^{1}(L, L′) = ⨁_{i′=1′}^{n′} Λ_{R}(L, A_{i′}) ⨁_{α′=1′}^{m′} Λ_{R}(L, B_{α′}).  (39)

An element X ∈ L can be represented as follows: X = x^{ia}e_{ia} + y^{α\overline{a}}e_{α\overline{a}}. Then the 1-forms

θ_{i′}^{ia} : L ∋ X ↦ x^{ia}e_{i′0} ∈ A_{i′}, θ_{α′}^{ia} : L ∋ X ↦ x^{ia}e_{α′0} ∈ B_{α′}  (40)

and

θ_{i′}^{α\overline{a}} : L ∋ X ↦ x^{α\overline{a}}e_{i′0} ∈ A_{i′}, θ_{α′}^{α\overline{a}} : L ∋ X ↦ x^{α\overline{a}}e_{α′0} ∈ B_{α′}  (41)

are generators of the A-module Λ_{R}^{1}(L, L′). The module Λ′_{A}^{1}(L, L′) = Hom′_{A}(L, L′) of all fibered A-linear L′-valued 1-forms is a submodule in Λ_{R}^{1}(L, L′) spanned by the 1-forms

θ_{i′}^{i} : L ∋ X ↦ X^{i}e_{i′0} ∈ A_{i′}, θ_{α′}^{i} : L ∋ X ↦ X^{i}e_{α′0} ∈ B_{α′}  (42)

and

θ_{α′}^{α} : L ∋ X ↦ Y^{α}e_{α′0} ∈ B_{α′}.  (43)

It is easily seen that Λ′_{A}^{1}(L, L′) is a direct summand in Λ_{R}^{1}(L, L′), that is, there exists a submodule \overline{Λ}_{R}^{1}(L, L′) in Λ_{R}^{1}(L, L′) such that

Λ_{R}^{1}(L, L′) = Λ′_{A}^{1}(L, L′) ⊕ \overline{Λ}_{R}^{1}(L, L′).  (44)

For example, one can take as \overline{Λ}_{R}^{1}(L, L′) the submodule spanned by the 1-forms

θ_{i′}^{iâ}, θ_{α′}^{ia^{∗}}, θ_{i′}^{α\overline{a}}, θ_{α′}^{αa^{∗}}.  (45)

Consider a foliated A-smooth manifold M^{L}. The tangent space T_{X}M^{L} of a foliated A-smooth manifold M^{L} at X ∈ M^{L} is an A-module isomorphic to L. Denote by L′ ⊗ Λ_{X}^{1}M^{L} = Λ_{R}^{1}(T_{X}M^{L}, L′) the A-module of L′-valued 1-forms on T_{X}M^{L} and by L′ ⊗ Λ^{1}M^{L} the fiber bundle of L′-valued 1-forms on M^{L}. L′ ⊗ Λ^{1}M^{L} is isomorphic to

⨁_{i′=1′}^{n′} A_{i′} ⊗ Λ^{1}M^{L} ⨁_{α′=1′}^{m′} B_{α′} ⊗ Λ^{1}M^{L}.  (46)

Using a smooth fiberwise A-linear metric in the bundle A ⊗ Λ^{1}M^{L}, one can decompose it into a Whitney sum of A-module bundles (cf. [22])

A ⊗ Λ^{1}M^{L} = Λ′(A)_{A−lin}^{1}M^{L} ⊕ Λ̃(A)^{1}M^{L} ,  (47)

where Λ′(A)_{X, A−lin}^{1}M^{L} is the A-module of fibered A-valued A-linear forms on T_{X}M^{L}. Here A is viewed as an A-module of the type A^{n′} ⊕ B^{m′} for n′ = 1 and m′ = 0. Since M^{L} is a foliated A-smooth manifold, the property of an A-linear form to be fibered does not depend on the choice of a chart from the foliated L-atlas on M^{L}. In terms of a local coordinates on M^{L}, the submodule Λ′(A)_{X,A−lin}^{1}M^{L} is spanned by the 1-forms dX^{i}, i = 1, …, n, and the submodule Λ̃(A)_{X}^{1}M^{L} by 1-forms of the following form

θ^{iâ} = dx^{iâ} + λ_{j}^{iâ}dX^{j},  θ^{α\overline{a}} = dy^{α\overline{a}} + λ_{j}^{α\overline{a}}dX^{j}.  (48)

Similarly, one gets a decomposition

B ⊗ Λ^{1}M^{L} = Λ′(B)_{A−lin}^{1}M^{L} ⊕ Λ̃(B)^{1}M^{L} ,  (49)

where the submodule Λ′(B)_{X, A−lin}^{1}M^{L} is spanned by the 1-forms dX^{i}, dY^{α}, i = 1, …, n, α = 1, …, m and the submodule Λ̃(B)_{X}^{1}M^{L} by 1-forms of the following form

θ′^{iâ} = dx^{iâ} + λ′_{j}^{iâ}dX^{j} + +λ′_{α}^{iâ}dY^{α},  θ′^{αa^{∗}} = dy^{αa^{∗}} + λ′_{α}^{αa^{∗}}dY^{α}.  (50)

The decompositions (47), (49) induce a decomposition

L′ ⊗ Λ^{1}M^{L} = Λ′(L′)_{A−lin}^{1}M^{L} ⊕ Λ̃(L′)^{1}M^{L} .  (51)

Denote by Λ(L′)_{X}^{r,s}M^{L} the submodule in L′ ⊗ Λ_{X}^{r+s}M^{L} spanned by the exterior products ξ_{1} ∧ ⋯ ∧ ξ_{r} ∧ η_{1} ∧ ⋯ ∧ η_{s}, where the 1-forms η_{1}, …, η_{s} are fibered Alinear and ξ_{1}, …, ξ_{r} ∈ Λ̃(L′)_{X}^{1}M^{L}. The union of the submodules Λ(L′)_{X}^{r,s}M^{L} for all X ∈ M^{L} is a subbundle

Λ(L′)^{r,s}M^{L} ⊂ L′ ⊗ Λ^{r+s}M^{L} → M^{L}.  (52)

Denote by Ω(L′)^{r,s}(U) the A-module of smooth sections of the bundle (52) over an open set U ⊂ M^{L}. From (48), (49) and (50) it follows that the exterior differential dω of a form ω ∈ Ω(L′)^{r,s}(U) belongs to

Ω(L′)^{r−1,s+2}(U) ⊕ Ω(L′)^{r,s+1}(U) ⊕ Ω(L′)^{r+1,s}(U) .  (53)

Assigning to a form ω the component d̂ω of the exterior differential dω belonging to Ω^{r+1,s}(U), we obtain the operator

d̂= d̂_{r,s} : Ω(L′)^{r,s}(U) → Ω(L′)^{r+1,s}(U).

Proposition 7. 1) d̂∘ d̂ = 0, and so, for each s, the following complex is associated to a foliated manifold M^{L}:

0 → Ω(L′)^{0,s}(M_{n}^{A}) →^{d̂} Ω(L′)^{1,s}(M_{n}^{A}) →^{d̂} Ω(L′)^{2,s}(M_{n}^{A}) →^{d̂} …. (54)

2) The homology groups H^{r,s}(L′)M^{L} of the complex (54) do not depend up to an isomorphism on the choice of a subbundle Λ̃(L′)^{1}M^{L}.

Proof. 1) Following [12], from (53) one can easily conclude that if ω ∈ Ω(L′)^{r,s}(U), then d^{2}ω ∈ Ω(L′)^{r−2,s+4}(U) ⊕ Ω(L′)^{r−1,s+3}(U) ⊕ Ω(L′)^{r,s+2}(U) ⊕ Ω(L′)^{r+1,s+1}(U) ⊕ Ω(L′)^{r+2,s}(U) and d̂^{2}ω is the component of d^{2}ω belonging to Ω(L′)^{r+2,s}(U). Therefore, d̂∘ d̂= 0.

2) Let Λ̃(L′)_{1}^{1}M_{n}^{A} and Λ̃(L′)_{2}^{1}M_{n}^{A} be two subbundles in the bundle L′⊗Λ^{1}M^{L} which are complementary to Λ′(L′)_{A−lin}^{1}M^{L}. The projection τ : Λ̃(L′)_{1}^{1}M^{L} → Λ̃(L′)_{2}^{1}M^{L} relative to Λ′(L′)_{A−lin}^{1}M^{L} is an A-module bundle isomorphism and induces isomorphisms

τ_{r,s} : Λ(L′)_{1}^{r,s}M^{L} → Λ(L′)_{2}^{r,s}M^{L}.

Let

τ′_{r,s} : Ω(L′)_{1}^{r,s}(M^{L}) → Ω(L′)_{2}^{r,s}(M^{L})

be the corresponding isomorphisms of A-modules of global sections of the bundles Λ(L′)_{1}^{r,s}M^{L} and Λ(L′)_{2}^{r,s}M^{L}. One can easily see that τ′_{r+1,s} ∘ d̂_{1} = d̂_{2} ∘ τ′_{r,s}. Consequently, the corresponding complexes (54) are isomorphic. □

The construction of d_{F} -cohomology of tensorial forms on a foliated principal bundle due to P.Molino [11], [12] can be extended to the case of tensorial (p, s)- forms on the bundle B_{A}M^{L} of fibered A-linear frames on M^{L} (see also [21], [24], where a similar cohomology was defined for A-smooth manifolds modelled on the A-module A^{n}).

Let ad be the adjoint representation of the Lie group GL′_{A}(L) in its Lie algebra gl′_{A}(L). A gl′_{A}(L)-valued r-form θ on B_{A}M^{L} is said to be tensorial [7] if R_{g}^{∗}θ = ad(g^{−1})θ for g ∈ GL′_{A}(L) and θ(v_{1}, …, v_{r}) = 0 when at least one of the vectors v_{1}, …, v_{r} is vertical. A tensorial gl′_{A}(L)-valued (r+s)-form θ is said to be an (r, s)-form if for any local section σ : U ⊂ M^{L} → B_{A}M^{L} the inverse image σ^{∗}θ is a local (r, s)-form on M^{L}. Let {σ_{κ} : U_{κ} ⊂ M^{L} → B_{A}M^{L}}_{κ∈A} be a family of foliated A-smooth sections such that {U_{κ}}_{κ∈A} is a covering of M^{L}. We have σ_{κ} = R_{g(κ,ν)} ∘ σ_{ν} where g(κ, ν) : U_{κ} ∩ U_{ν} → GL′_{A}(L) are foliated A-smooth functions. The forms θ_{κ} = σ_{κ}^{∗}θ are related as follows:

θ_{κ} = ad(g(κ, ν)^{−1})θ_{ν}.  (55)

The tensorial gl′_{A}(L)-valued (r, s)-forms on B_{A}M^{L} are in one-to-one correspondence with the families {θ_{κ}}_{κ∈A} of local gl′_{A}(L)-valued (r, s)-forms on M^{L} satisfying the relations (55). Since ad(g(κ, ν)^{−1}) are foliated A-smooth functions, from (55) it follows that d̂θ_{κ} = ad(g(κ, ν)^{−1})d̂θ_{ν}. Consequently, the family {d̂θ_{κ}}_{κ∈A} determines a tensorial (r + 1, s)-form. Denote it by d̂θ. Thus, there arises an operator

d̂: Ω_{ad}^{r,s}(B_{A}M^{L}, gl′_{A}(L)) → Ω_{ad}^{p+1,s}(B_{A}M^{L}, gl′_{A}(L)),

where Ω_{ad}^{r,s}(B_{A}M^{L}, gl′_{A}(L)) denotes the A-module of gl′_{A}(L)-valued tensorial (r, s)-forms on B_{A}M^{L}. Denote by H_{ad}^{r,s}(B_{A}M^{L}, gl′_{A}(L)) the corresponding cohomology groups.

A connection Γ in the bundle B_{A}M^{L} with connection form

ω : T (B_{A}M^{L}) → gl′_{A}(L)  (56)

is called adapted if ω_{Z} : T_{Z}(B_{A}M^{L}) → gl′_{A}(L) is a fibered A-linear mapping for each Z ∈ B_{A}M^{L}. A connection Γ is called foliated A-smooth if the mapping (56) is foliated A-smooth.

Suppose, an adapted connection Γ in the bundle (B_{A}M^{L}) is given and Ω is the curvature form of the connection Γ. Then ([7], Ch. II, §5)

dω(u, v) = −\frac{1}{2}[ω(u), ω(v)] + Ω(u, v), u, v ∈ T_{Z}(B_{A}M^{L}).  (57)

For any local foliated A-smooth section σ_{κ} : U_{κ} → B_{A}M^{L}, the form σ_{κ}^{∗}ω is fibered A-linear, and the form σ_{κ}^{∗}([ω, ω]) belongs to Ω_{ad}^{0,2}(M^{L}, gl′_{A}(L)), and σ_{κ}^{∗}dω = dσ_{κ}^{∗}ω ∈ Ω_{ad}^{0,2}(M^{L}, gl′_{A}(L)) + Ω_{ad}^{1,1}(M^{L}, gl′_{A}(L)). Consequently, the (1, 1)-component Ω^{1,1} of the curvature form of an adapted connection satisfies the relations σ_{κ}^{∗}Ω^{1,1} = d̂s_{κ}^{∗}ω. Hence it follows that σ_{κ}^{∗}(d̂Ω^{1,1}) = 0. Thus, the form Ω^{1,1} is d̂-closed, and the class

a(B_{A}M^{L}) = [Ω^{1,1}] ∈ H_{ad}^{1,1}(B_{A}M^{L}, gl′_{A}(L))

74 V. V. SHURYGIN, L. B. SMOLYAKOVA

is well-defined. This class does not depend on the choice of an adapted connection ω. In fact, suppose that ω′ is the connection form of another adapted connection in B_{A}M^{L}. Then [12] ω′ − ω = τ is a tensorial form belonging to Ω_{ad}^{0,1}(B_{A}M^{L}, gl′_{A}(L)). Therefore Ω′ ^{1,1} − Ω^{1,1} = d̂τ. The class a(B_{A}M^{L}) is called (cf. [2], [21]) the Atiyah-Molino class of M^{L}.

Similarly to Theorem 2.1 of [12], the following theorem holds. Theorem 10. a(B_{A}M^{L}) = 0 if and only if there exists a foliated A-smooth

connection in the bundle B_{A}M^{L}.

Proof. (cf. [12]) If ω is a foliated A-smooth form, then for any local foliated A-smooth section σ of the bundle B_{A}M^{L} we have σ^{∗}ω ∈ Ω_{ad}^{0,2}(M^{L}, gl′_{A}(L)), and so a(B_{A}M^{L}) = 0.

Conversely, if [Ω^{1,1}] = 0, then Ω^{1,1} = d̂τ, and the connection Γ′ in B_{A}M^{L} determined by the connection form ω′ = ω − τ is a foliated A-smooth connection because d̂σ_{κ}^{∗}ω′ = d̂σ_{κ}^{∗}ω − d̂σ_{κ}^{∗}τ = σ_{κ}^{∗}Ω^{1,1} − σ_{κ}^{∗}d̂τ = 0. □

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Department of Mathematics, Kazan State University, Kremlevskaya 18, 420008, Kazan, Russia E-mail address: Vadim.Shurygin@ksu.ru E-mail address: larissa@freemail.ru

Received December, 1, 2001