Lobachevskii Journal of Mathematics Vol. 5, 1999, 29–55 \begin{matrix}c \\ ◯ \end{matrix}Vadim V. Shurygin

Vadim V. Shurygin

THE STRUCTURE OF SMOOTH MAPPINGS OVER WEIL ALGEBRAS AND THE CATEGORY OF MANIFOLDS OVER

ALGEBRAS

(submitted by B.N.Shapukov)

Abstract. As is known, the bundle T^{A}M_{n} of infinitely near points of A-type defined for any local Weil algebra A and smooth real manifold M_{n} is one of basic examples of smooth manifolds over A. In the present paper we give a description of the local structure of smooth mappings in the category of smooth manifolds over local algebras and consider various examples of such manifolds. Next we study the homotopy and holonomy groupoids of a smooth manifold M_{n}^{A} over a local algebra A associated with canonical foliations corresponding to ideals of A. In particular, it is proved that a complete manifold M_{n}^{A} has neither homotopy nor holonomy vanishing cycles.

1. Introduction

A mapping f : U ⊂ L → U′ ⊂ L′ between domains of modules L and L′ over a commutative associative algebra A is said to be A-smooth if the tangent mappings T_{x}f, x ∈ L, are A-linear. A manifold over algebra A is a smooth manifold with an atlas whose charts take values in a fixed Amodule L and coordinate transformations are A-smooth. The general theory of manifolds over algebras was studied by A.P.Shirokov [34], V.V. Vishnevsky [46], G.I.Kruchkovich [19], J. Vanžura [45], and other researchers. A.P.Shirokov discovered that structures of finite-dimensional manifolds over local algebras arise on tangent bundles and bundles of infinitely near points in the sense of A.Weil [35], [36]. V.V.Vishnevsky [46] introduced semitangent bundles over towers of submersions and studied the structures of manifolds over algebras of plural numbers which arise on such bundles.

Various types of manifolds over algebras can also be defined in the case of infinite dimensional manifolds and algebras. S. Kobayashi [13] studied manifolds modelled on modules over algebras of smooth functions on a compact manifold. V.S.Vladimirov and I.V.Volovich [47] studied differentiable functions over commutative Banach superalgebras. Structures of smooth manifolds over limits of projective spectra of finite-dimensional algebras arise on the infinite dimensional jet manifolds considered by I.N.Bernstein and B.I.Rosenfeld. Weil functors on the category of infinite dimensional manifolds modelled on convenient locally convex vector spaces were studied by A. Kriegl and P.W. Michor [18]. Another generalization of Weil functors for infinite dimensional manifolds was constructed by I. Kolář [16]. M.H. Papatriantafillou [27], [28] studied manifolds modelled on modules over commutative locally multiplicatively convex algebras.

In the present paper we describe the local structure of smooth mappings in the category of manifolds over local algebras in the sense of A.Weil [40], consider examples of manifolds over local algebras and study homotopy and holonomy groupoids of such manifolds.

§2 contains a brief description of the category of manifolds modelled on modules over commutative associative algebras.

In §3, we state in the most general form the theorem on local structure of an A-smooth mapping from an open subset of the module A^{n} = A × ⋯ × A of n-tuples of elements of a local algebra A to the A-module A^{k} (Theorem 1) and give the complete proof of this theorem. Some variants of Theorem 1 were given in [37] and [40]. Next, in the case when a local algebra A is a semidirect sum A = \overline{A}⊕I of a subalgebra \overline{A} and an ideal I, we establish the expression of an A-smooth mapping from an open subset \overline{U} of the A-module \overline{A}^{n} to the Amodule A^{k} and prove that such a mapping can be prolonged to an A-smooth mapping between domains of A-modules A^{n} and A^{k} (Theorem 2). This result allows to construct a functor from the category of finite-dimensional \overline{A}-smooth manifolds to the category of finite-dimensional A-smooth manifolds which is a generalization of the Weil functor of A-prolongation (see Example 8).

Theorem 3, which was proved in [39] (see also [40]), establishes the local expression of an A-smooth mapping between A-modules of the form A^{n} ⊕R^{m}. Structures of A-smooth manifolds modelled on modules of such type arise on transverse Weil bundles [39], [40]. Here we give the proof of this theorem and correct an inaccuracy in the statement of the theorem in [39].

In §4, we consider examples of manifolds over algebras and functors on categories of manifolds over algebras. In particular, we construct a generalization of the Weil A-prolongation functor in the case when A is a semidirect sum A = \overline{A} ⊕ I.

In §5, we define the canonical osculating fiber bundle O_{I}^{V} (M_{n}^{A}) and the homotopy groupoid Π_{I}M_{n}^{A} associated with the canonical I^{n}-foliation F^{I} induced by an ideal I of a local algebra A on an n-dimensional A-smooth manifold M_{n}^{A}. It is shown that O_{I}^{V} (M_{n}^{A}) and Π_{I}M_{n}^{A} carry natural structures of smooth manifolds over the algebra Â_{I} which is obtained by doubling the ideal I in A.

In this section we also define the developing map D_{I} : Π_{I}(M_{n}^{A}) → O_{I}^{V} (M_{n}^{A}) and prove (basing on results of P. Dazord and G. Hector [6]) that the canonical I^{n}-foliation F^{I} on a complete A-smooth manifold M_{n}^{A} has no vanishing cycles.

In §6, we associate with each leaf of the canonical I^{n}-foliation on M_{n}^{A} the I-holonomy representation and define the I-holonomy groupoid Γ_{I}M_{n}^{A} of M_{n}^{A}. The I-holonomy groupoid Γ_{I}M_{n}^{A} also carries a natural structure of smooth manifold over the algebra Â_{I}. Following R. Wolak [53], we introduce the notion of an I-holonomy vanishing cycle for a manifold M_{n}^{A} and prove that M_{n}^{A} has no I-holonomy vanishing cycles if and only if Γ_{I}(M_{n}^{A}) is a Hausdorff topological space.

2. Preliminaries

Let A denote the category of associative commutative finite-dimensional Ralgebras and A−Mod the category of modules over algebras belonging to A which are finite-dimensional vector spaces over R. An object of A−Mod is a triple (A, L, ∗), where A is an object of A, L is a finite-dimensional real vector space, and ∗ : A × L → L a bilinear mapping endowing L with a structure of A-module: ∗(α, X) = αX. For the sake of brevity, we shall also denote a module (A, L, ∗) by (A, L) or L when this does not cause ambiguities. By a morphism in A−Mod from (A, L) to (A′, L′) we mean a pair (φ, ψ) consisting of a morphism of algebras φ : A → A′ and a linear mapping ψ : L → L′ satisfying the relation ψ(αX) = φ(α)ψ(X). The composition of morphisms (φ′, ψ′) and (φ, ψ) in A−Mod is the pair of compositions (φ′, ψ′) ∘ (φ, ψ) = (φ′ ∘ φ, ψ′ ∘ ψ).

The category A−Man of smooth manifolds over algebras is described as follows. Let M be a smooth manifold and (A, L) a module of the category A−Mod. An L-chart on M is a pair (U, h) consisting of an open set U ∈ M and a diffeomorphism h : U → U^{∗} ⊂ L. An L-atlas on M is a collection of L-charts {(U_{α}, h_{α})}_{α∈A} such that {U_{α}}_{α∈A} is a covering of M and the pair consisting of the identity isomorphism id_{A} : A → A and the tangent mapping

T_{h_{β}(x)}(h_{α} ∘ h_{β}^{−1}) : T_{h_{β}(x)}L ≡ L → L ≡ T_{h_{α}(x)} ,  (1)

is an isomorphism of A-modules for all x ∈ M, α, β ∈ A. An object of the category A−Man is a quadruple (A, L, M, Φ), where Φ is a maximal L-atlas on M. An object (A, L, M, Φ) is called an L-manifold or an A-smooth manifold modelled on L. Briefly, an object (A, L, M, Φ) is denoted by (A, L, M), (L, M), or simply M when this does not lead to ambiguities.

The isomorphism (1) allows to transfer by means of T_{x}h_{α}^{−1} the structure of an A-module from T_{x}U_{α}^{∗} ≅ L to the tangent space T_{x}M of an L-manifold M at x ∈ U_{α}. Since T h_{α}^{−1} : T U_{α}^{∗} → T U_{α} is a smooth mapping, there arises a smooth action

θ : A × T M ∋ (α, v_{x}) ↦ αv_{x} ∈ T M ,  (2)

which induces a polyaffinor structure on M, a representation of A in the space of smooth (1,1)-tensor fields on M [19].

A morphism from (A, L, M) to (A′, L′, M′) is a pair (φ, f) consisting of a morphism of algebras φ : A → A′ and a smooth mapping f : M → M′ such that (φ, T_{x}f) is a morphism of modules for all x ∈ M. Let {e_{a}}, {e_{a′}}, {E_{i}}, and {E_{i′}} be bases in the algebras A , A′ and modules L, L′ viewed as real vector spaces, φ_{a}^{a′}, γ_{aj}^{i}, and γ_{a′j′}^{i′} the components of the morphism φ and bilinear mappings ∗ : A × L → L and ∗′ : A′ × L′ → L′ respectively. In terms of local coordinates {x^{i}} and {x^{i′}} on the manifolds M and M′ which correspond, respectively, to the bases {E_{i}} and {E_{i′}}, the condition for a pair (φ, f) to be a morphism in A−Man can be written as follows (we adopt Einstein’s summation convention):

\frac{∂f^{i′}}{∂x^{k}} γ_{aj}^{k} = φ_{a}^{a′}γ_{a′k′}^{i′} \frac{∂f^{k′}}{∂x^{j}} .  (3)

Fixing an algebra A in A and the identity mapping id_{A} as the only morphism from A to A, we get, respectively, the subcategories A−Mod in A−Mod consisting of A-modules and A−Man in A−Man consisting of manifolds over A. Morphisms in A−Man are called A-smooth mappings, and isomorphisms, respectively, A-diffeomorphisms. Any morphism φ : A′ → A generates a functor (denote it by the same letter) φ : A−Mod → A′−Mod (see [30], §2.1) that assigns to an A-module (A, L) the A′-module (A′, L) in which the action of A′ on L is defined by the rule α′X = φ(α′)X and to a morphism ψ : L → L′ in A−Mod the same mapping ψ viewed as a morphism in A′−Mod. A morphism φ : A′ → A generates also a functor φ : A−Man → A′−Man which assignes to a manifold (A, L, M, Φ) the manifold (A′, L, M, Φ). In particular (see [30], §2.1), if i : A′ → A is an inclusion of a subalgebra and p : A′ → A = A′/ker p is the natural ephimorphism, then any A-smooth manifold can also be viewed as an A′-smooth manifold.

Let M be an A-smooth manifold modelled on an A-module L. Denote by GL(L, A) the group of automorphisms of L. Let L′ ⊂ L be a submodule invariant under the action of GL(L, A) on L. For any x ∈ M, let us choose an L-chart (U_{α}, h_{α}) on M such that x ∈ U_{α} and consider the subspace L′_{x} in T_{x}M which is the image of the submodule L′ ⊂ L ≅ T_{h_{α}(x)}L under the mapping T_{x}h_{α}^{−1}. Since the submodule L′ ⊂ L is invariant under the action of GL(L, A), the subspace L′_{x} does not depend on the choice of an L-chart (U_{α}, h_{α}). The correspondence M ∋ x ↦ L′_{x} ⊂ T_{x}M ⊂ T M is a smooth completely integrable distribution on M, and each maximal integral manifold M′ of this distribution carries an induced structure of L′-manifold such that the inclusion ι : M′ → M is an A-smooth mapping [40]. The corresponding foliation on M is called the canonical L′-foliation. Leaves of the canonical L′-foliation on an L-manifold M are called L′-submanifolds.

Since any algebra can be embedded as a subalgebra into a unital algebra (an algebra with unit), any manifold in A−Man can be viewed as a manifold over a unital algebra.

In what follows, we suppose that all algebras under consideration are unital. The unit 1_{A} of an algebra A will also be denoted by δ. In accordance with this, the expansion of 1_{A} in terms of a basis {e_{a}} will be written as follows: 1_{A} = δ^{a}e_{a}.

The equations (3) are a generalization of Scheffers’ conditions [33], [34] for A-differentiability of a function f : U ⊂ A → A:

∂_{c}f^{b}γ_{ag}^{c} = γ_{ac}^{b}∂_{g}f^{c},  (4)

where γ_{ac}^{b} are the structure constants of A with respect to a basis {e_{a}}, and ∂_{a}f^{b} = ∂f^{b}/∂x^{a}. Scheffers’ conditions (4) can also be written in the form [34]:

∂_{a}f^{b} = γ_{ac}^{b}δ^{g}∂_{g}f^{c}.  (5)

For a function of several variables f : U ⊂ A^{n} → A, f : {X^{i} = x^{ia}e_{a}} ↦ f(X^{i}) = f^{b}(x^{ia})e_{b}, where A^{n} = A × ⋯ × A is the A-module of n-tuples of elements of A, Scheffers’ conditions of A-differentiability are of the form

∂_{ia}f^{b} = γ_{ac}^{b}δ^{g}∂_{ig}f^{c}.  (6)

If a function f satisfies the equations (6), its differential can be represented in the form df = f_{i}dX^{i}, where f_{i} = δ^{a}∂_{ia}f is the partial derivative with respect to the variable X^{i} ∈ A. The functions f_{i}(X^{j}), i = 1, …, n, are also A-differentiable.

3. Smooth functions over local Weil algebras

A unital algebra A ∈ A is said to be local in the sense of A.Weil [50], [54],

[17] if its radical (the set of 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^{∘} .  (7)

The structure of a local algebra A is completely determined by its radical, the nilpotent algebra A^{∘}. The theory of nilpotent algebras was elaborated in details by V.V.Wagner in [48], [49] in connection with the study of differential groups and tangent spaces of higher order. A detailed exposition of the theory of local Weil algebras can also be found in the book of I. Kolář, P.W. Michor, and J. Slovák [17].

The ideal A^{∘} is generated by the collection of elements {τ^{A}}, A = 1, …, N,

such that the collection of residue classes {τ^{A} + (A^{∘})^{2}}, where (A^{∘})^{2} denotes

the square of the ideal A^{∘}, is a basis of the quotient algebra A^{∘}/(A^{∘})^{2}. The

collection {τ^{A}}, A = 1, …, N, is called a pseudobasis in A^{∘} [49], and the

number N = dim_{R}A^{∘}/(A^{∘})^{2} 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 hight of A. Let R[[N]] = R[[t^{1}, …, t^{N}]] be the algebra of formal power series in N

variables t^{1}, …, t^{N} with coefficients in R, R^{∘}[[N]] the maximal ideal of R[[N]], which consists of series with zero constant term. The mapping φ which assigns to a formal series α = ∑_{|p|=0}^{∞} α_{p}t^{p}, where p = (p_{1} …, p_{N}) is a multiindex, |p| = p_{1} + ⋯ + p_{N}, the element ∑_{|p|=0}^{∞} α_{p}τ^{p} ∈ A is a homomorphism of algebras and A ≅ R[[N]]/ker φ. In particular, the algebra R(N, q) of truncated polynomials of degree ≤ q in N variables is isomorphic to the quotient algebra R[[N]]/R^{∘}[[N]]^{q+1}. The algebra R(1, 1) is denoted by R(ε) and called the algebra of dual numbers. In [17] this algebra is denoted by D. Let ε^{A} ∈ R(N, q) be the residue class containing the element t^{A} ∈ R[[N]]. The products ε^{p} = (ε^{1})^{p_{1}} … (ε^{N})^{p_{N}} for 0 ≤ |p| ≤ q constitute a basis in R(N, q). The linear mapping ψ : R(N, q) → A which maps ε^{p} to τ^{p} is an epimorphism of algebras and A ≅ R(N, q)/ker ψ.

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

A/(A^{∘})^{r+1}, in particular, the epimorphism π_{r}^{q} : R(N, q) → R(N, r). For r = 0,

π_{0}^{q} : A → R = A/A^{∘} is the epimorphism which assigns to an element X ∈ A its real part x ∈ R. The epimorphism π_{r}^{q} induces the epimorphism of modules π_{r}^{q} : A^{n} → A_{(r)}^{n}, which is denoted by the same symbol.

Using a Jordan-Hölder composition series [30]

A ⊃ A^{∘} = I_{1} ⊃ I_{2} ⊃ … ⊃ I_{m} ⊃ 0,  (8)

where I_{a}/I_{a+1} is one-dimensional algebra with zero multiplication, one can choose a basis

{e_{a}} = {e_{0}, eâ}, a = 0, 1, …, m = dim A^{∘}, â = 1, …, m,  (9)

in A such that e_{0} = 1 ∈ R, eâ ∈ Iâ, eâ ∉ I_{â+1}. The expansion of X ∈ A in terms of the basis (9) will be written as follows: X = x^{a}e_{a} = x^{0} + xâeâ.

The structure constants of A with respect to the basis (9) satisfy the following relations: γ_{0a}^{b} = δ_{a}^{b} (Kronecker’s deltas), γ_{ab̂}^{c} = 0 when a ≥ c, and so Scheffers’ equations (6) take the form

∂_{ia}f^{b} = γ_{ag}^{b}∂_{i0}f^{g}.  (10)

Let I ⊂ A be an ideal, \overline{A} = A/I the quotient algebra, and p : A → \overline{A} the natural epimorphism. The epimorphism p induces the epimorphism of modules of n-tuples p : A^{n} → \overline{A}^{n} (for the sake of brevity, we use the same symbol for designation of this epimorphism). The foliation on A^{n} determined by the submersion p is called the canonical I^{n}-foliation. 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 of a foliated manifold (M, F) is called projectable or basic (see, for example, [25]) if f is constant along leaves of the foliation F.

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

mapping 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}^{∘},  (11)

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

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 (11) 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 there holds φ = (Φ̃|π_{0}^{q}(U)) ∘ π_{0}^{q} .

Proof. To prove the first statement, it suffices to verify that the formula (11) determines an A-smooth mapping for k = 1. By definition, a function Φ : U ∋ {X^{i}} ↦ Y = Φ(X) ∈ A is an A-smooth if its differential dΦ is an A-linear 1-form on U, which is equivalent to the fact that dΦ = Φ_{i}dX^{i}. Let

Φ(X^{i}) = ∑_{|p|=0}^{q} \frac{1}{p!} \frac{D^{p}φ}{Dx^{p}} X^{p}^{∘},  (12)

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

X^{∘}^{0} = 1. We have

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

∑_{i=1}^{n} ( ∑_{|p|=0}^{q} \frac{1}{p!} \frac{D^{p}}{Dx^{p}} (\frac{∂φ}{∂x^{i}} ) X^{p}^{∘}) dx^{i} + ∑_{i=1}^{n} ( ∑_{|w|=0}^{q−1} \frac{1}{w!} \frac{D^{w}}{Dx^{w}} (\frac{∂φ}{∂x^{i}} ) X^{∘}^{w}) dX^{∘}^{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. The upper limit of summation over w in the second summond in the last expression can be replaced by q (an additional

term corresponding to w = q is zero since dX^{∘}^{i} ∈ A^{∘}). As a result, we obtain

dΦ = ∑_{i=1}^{n} ( ∑_{|p|=0}^{q} \frac{1}{p!} \frac{D^{p}}{Dx^{p}} (\frac{∂φ}{∂x^{i}} ) X^{p}^{∘}) dX^{i}.  (13)

To prove the second statement, let us assume first that the domain U is an open coordinate parallelepiped determined by the system of inequalities α^{ia} < x^{ia} < β^{ia}, where α^{ia} < 0 and β^{ia} > 0. Let us denote φ = Φ|R^{n} ∩ U and show that there exist a unique A-smooth function Ψ : U → A such that

Ψ|R^{n} ∩ U = φ.  (14)

The function Ψ(X^{i}) = ψ^{a}(x^{ib})e_{a} satisfies Scheffers’ equations (10)

∂_{ia}ψ^{b} = γ_{ag}^{b}∂_{i0}ψ^{g}.  (15)

Putting b = 0 in (15), we obtain the relations ∂_{i0}ψ^{0} = ∂_{i0}ψ^{0} and ∂_{ia}ψ^{0} = 0 for a > 0, hence it follows that ψ^{0} depends only on x^{i0}, therefore, ψ^{0}(x^{i0}) = φ^{0}(x^{i0}). Next, fixing b > 0 in (15), we obtain the following system of equations for the function ψ^{b}(x^{ia}):

∂_{ia}ψ^{b} = ∑_{g=2}^{b−1} γ_{ag}^{b}∂_{i0}ψ^{g} for b > a > 0,

∂_{ib}ψ^{b} = ∂_{i0}ψ^{0}, ∂_{ia}ψ^{b} = 0 for a > b.

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

∂_{i1}ψ^{1} = ∂_{i0}ψ^{0}, ∂_{ia}ψ^{1} = 0 for a > 1.

Taking into account (14), we obtain

ψ^{1} = x^{i1}∂_{i0}φ^{0} + φ^{1}.  (16)

Taking the formula (16) as the basis for mathematical induction, let us show that there exists a unique function ψ^{b} (b = 2, …, m) satisfying (14) and Scheffers’ conditions, and that ψ^{b} is a polynomial in x^{ia}, a ≤ b, with coefficients depending on x^{i0}. Assuming that this statement holds for all a < b, we obtain the following system of equations for ψ^{b}:

∂_{ia}ψ^{b} = P_{ia}, a ≤ b,  (17)

where P_{ia} are polynomials in x^{jg}, 0 < g < b, with coefficient depending on x^{j0}. It can be verified by direct calculations that the integrability conditions of the system (17) are fulfilled: ∂_{jc}P_{ia} = ∂_{ia}P_{jc}. But this fact also follows from the fact that the function

Ψ(X^{i}) = ∑_{|p|=0}^{q} \frac{1}{p!} \frac{D^{p}φ}{Dx^{p}} X^{p}^{∘}  (18)

satisfies the equations (15). The solution of the system (17) is a polynomial in x^{ia}, 0 < a ≤ b, and is determined up to a summond depending on x^{i0}. The requirement ψ^{b}|R^{n} ∩ U = φ^{b} determines the function ψ^{b} uniquely. Thus, there exists only one A-smooth function Ψ(X^{i}) on U satisfying the relation (14). Hence it follows that Φ(X^{i}) coincides with the function (18).

Let now the domain U be an arbitrary coordinate parallelepiped α^{ia} < x^{ia} < β^{ia} and X_{0} ∈ U. The translation λ : A^{n} ∋ Y ↦ X = Y + X^{∘}_{0} ∈ A^{n} is an A-diffeomorphism. Consider W = λ^{−1}(U). The composition Ψ = Φ ∘ λ : W → A is an A-smooth function. It is uniquely determined by the restriction ψ = (Φ ∘ λ)|R^{n} ∩ W and is of the form

Ψ(Y^{i}) = ∑_{|p|=0}^{q} \frac{1}{p!} \frac{D^{p}ψ}{Dx^{p}} Y^{∘} ^{p}.  (19)

Hence it follows that the function Φ = Ψ ∘ λ^{−1} is of the form

Φ(X^{i}) = ∑_{|p|=0}^{q} \frac{1}{p!} \frac{D^{p}ψ}{Dx^{p}} (X^{∘} −X^{∘}_{0})^{p} = ∑_{|p|=0}^{q} \frac{1}{p!} \frac{D^{p}φ}{Dx^{p}} X^{p}^{∘},  (20)

where

φ(x^{i}) = ∑_{|p|=0}^{q} \frac{1}{p!} \frac{D^{p}ψ}{Dx^{p}} (−X^{∘}_{0})^{p}.  (21)

The uniqueness of the representation (20) follows from that of the representation (19).

Let now U ⊂ A^{n} be an arbitrary open subset, X ∈ U, W_{X} ⊂ U an open coordinate parallelepiped containing X, and φ_{W_{X}} the function given by the formula (21) for the domain W_{X}. Then an A-smooth function Φ : U → A is of the form (12), where the basic function φ is defined as follows: φ(X) = φ_{W_{X}}(π_{0}^{q}(X)). The fact that φ(X) is well-defined follows from the uniqueness of the representation of the function Φ in the form (20) in a neighborhood of X.

The third statement of the theorem follows immediately from the first two ones. □

Definition. The mapping Φ : U → A^{k} given by the formulas (11) is called the analytic prolongation of the projectable mapping φ : U → A^{k}.

The analytic prolongation of a mapping φ will be denoted by φ^{A}.

Proposition 1. There hold the following properties of analytic prolongations:

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

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

4^{∘}. (D^{p}φ/Dx^{p})^{A} = D^{p}φ^{A}/DX^{p} for φ : U ⊂ A^{n} → A.

Proof. The above indicated properties follow immediately from (11), (13), and the uniqueness of the representation of an A-smooth mapping in the form (11). □

Let p : A ∋ X ↦ \overline{X} ∈ \overline{A} = A/I be an epimorphism of local algebras, p : A^{n} → \overline{A}^{n} the corresponding epimorphism of modules, and \overline{U} ⊂ \overline{A}^{n}, U ⊂ A^{n} arbitrary open sets.

Proposition 2. 1) A mapping Φ : p^{−1}(\overline{U}) → \overline{A}^{k} is A-smooth if and only if it is of the form Φ = \overline{Φ} ∘ p, where \overline{Φ} : \overline{U} → \overline{A}^{k} is an \overline{A}-smooth mapping. An A-smooth mapping Φ : U → \overline{A}^{k} is projectable with respect to the canonical I^{n}-foliation on U.

2) Let a mapping φ : U → \overline{A}^{k} be projectable with respect to the canonical

A^{n}^{∘}-foliation on U. Then the mapping Φ : U → \overline{A}^{k} determined by the formula

\overline{Y} ^{i′} = Φ^{i′}(X) = ∑_{|p|=0}^{q} \frac{1}{p!} \frac{D^{p}φ^{i′}}{Dx^{p}} \overline{X}^{∘}^{p},  (22)

is A-smooth, and any A-smooth mapping Φ : U → \overline{A}^{k} is of the form (22).

3) An \overline{A}-smooth mapping \overline{Φ} : \overline{U} → \overline{A}^{k} can be prolonged to an A-smooth mapping Φ̃ : p^{−1}(\overline{U}) → A^{k}.

Proof. Identifying T_{X}U ≡ A^{n}, X ∈ U, from the conditions of A-smoothness for the mapping Φ we have d_{X}Φ|I^{n} = 0, which is equivalent to the fact that Φ = \overline{Φ} ∘ p. Now the first statement follows from the fact that dΦ is A-linear with respect to p if and only if d\overline{Φ} is \overline{A}-linear.

The second statement is a consequence of the first one and Theorem 1. Let \overline{Φ} : \overline{U} → \overline{A}^{k} be an \overline{A}-smooth mapping. Then the mapping Φ = \overline{Φ} ∘ p :

p^{−1}(\overline{U}) → \overline{A}^{k} is A-smooth and, therefore, it is of the form

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

Consider an arbitrary linear mapping λ : \overline{A}^{k} → A^{k} such that p ∘ λ = id and let φ̃ = λ ∘ φ : p^{−1}(\overline{U}) → A^{k}. Then the mapping Φ̃ given by the equations

Φ̃^{i′}(X) = ∑_{|p|=0}^{q} \frac{1}{p!} \frac{D^{p}φ̃^{i′}}{Dx^{p}} X^{∘}^{p}

satisfies the conditions of the third statement. □

Consider now the case when a local algebra A is a semidirect sum A = \overline{A}⊕I of a subalgebra \overline{A} ≅ A/I and an ideal I. The tensor product of two local algebras provides an example of such situation:

A′ ⊗A = A′ ⊗(R⊕A^{∘}) = A′ ⊕(A′⊗A^{∘}).

Let i : \overline{A} → A be the inclusion and p : A → \overline{A} the canonical epimorphism. The same symbols will be used for the corresponding mappings of the modules of n-tuples. Since \overline{A} is a subalgebra in A, any A-module is also an \overline{A}-module. In particular, we shall consider the module A^{n} as an \overline{A}-module.

Theorem 2. Let a local algebra A be a semidirect sum A = \overline{A} ⊕ I of a subalgebra \overline{A} and an ideal I. Then:

1) An \overline{A}-smooth mapping Φ : \overline{U} ⊂ \overline{A}^{n} → A^{k} is of the form

Y^{i′} = φ^{i′} + ∑_{|p|=1}^{q} \frac{1}{p!} \frac{D^{p}φ^{i′}}{Dx^{p}} \overline{X}^{∘}^{p}, \overline{X}^{∘}^{i} ∈ \overline{A}^{∘},  (23)

where φ^{i′} are basic functions with respect to the canonical \overline{A}^{∘} ^{n}-foliation.

2) An \overline{A}-smooth mapping Φ : \overline{U} ⊂ \overline{A}^{n} → A^{k} can be uniquely prolonged to an A-smooth mapping Φ̃ : p^{−1}(\overline{U}) ⊂ A^{n} → A^{k} such that Φ̃|\overline{U} = Φ. If Φ is given by the equations (23), then Φ̃ is of the form

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

Proof. Consider a Jordan-Hölder composition series (8) containing the ideal I = I_{k}:

A ⊃ A^{∘} = I_{1} ⊃ I_{2} ⊃ … ⊃ I_{k} ⊃ … ⊃ I_{m} ⊃ 0.

Let the indices \overline{a}, \overline{b}, … run from 0 to k − 1, and the indices ǎ, b̌, … from k to m. Let us choose a basis (9) \{e_{a} = e\overline{a}, eǎ\} in A in such a way that e\overline{a} ∈ \overline{A} ∩ I\overline{a}. Then the elements \{e\overline{a}\} constitute a basis of \overline{A}, and \{eǎ\} a basis of I. In terms of the chosen basis \{e_{a} = e\overline{a}, eǎ\}, Scheffers’ conditions of \overline{A}-smoothness of a mapping Φ : \overline{U} ⊂ \overline{A}^{n} → A^{k} are of the form

γ\overline{a} ; \overline{b} \overline{c}∂_{i \overline{a}}Φ^{i′ g} = γ_{\overline{b} a}^{g}∂_{i \overline{c}}Φ^{i′ a},  (25)

where γ_{bc}^{a} are the structure constants of A. Putting \overline{c} = 0 in (25), we obtain the following equations, which are similar to (15):

∂_{i \overline{b}}Φ^{i′ g} = γ_{\overline{b} a}^{g}∂_{i0}Φ^{i′ a}.  (26)

Then, in the same way as in the proof of Theorem 1, one can show that formulas (23) determine an \overline{A}-smooth mapping, and that from (26) it follows that the representation (23) is unique for given basic functions φ^{i′}.

The second statement follows from the fact that each of the mappings (23) and (24) is uniquely determined by the basic functions φ^{i′}. □

The next theorem establishes the local structure of A-smooth mappings between A-modules of more general form A^{n} ⊕ R^{m}. Structures of A-smooth manifolds modelled on modules of such type arise on bundles of transverse A-velocities [40]. The structure of an A-module on the direct sum of vector spaces A^{n} ⊕ R^{m} is defined as follows:

∗ :A^{∘} ×(A^{n} ⊕ R^{m}) ∋ (α, u ⊕ v) ↦ αu ⊕ 0 ∈ A^{n} ⊕ R^{m}.  (27)

To formulate the theorem, let us choose in A a basis of the following special form: \{e_{0} = 1, e\overline{a}, eâ\} = \{e_{a}\}, where \{e\overline{a}, eâ\} is a basis in A^{∘}, and \{eâ\} a basis

in the ideal Ann(A^{∘}) annihilating A^{∘}. Then an arbitrary element of A^{n} ⊕ R^{m} is of the form \{X^{i} = x^{i} + x^{i\overline{a}}e\overline{a} + x^{iâ}eâ, y^{u}\}, and a mapping

Φ : W ⊂ A^{n} ⊕ R^{m} → A^{n′} ⊕ R^{m′}

can be written as follows:

Φ : W ∋ {X^{i}, y^{u}} ↦ {Φ^{i′}(X^{i}, y^{u}), φ^{u′}(X^{i}, y^{u})} ∈ A^{n′} ⊕ R^{m′}.  (28)

Theorem 3. Let W ⊂ A^{n} ⊕ R^{m} be an open coordinate parallelepiped in A^{n} ⊕ R^{m}. Then an A-smooth mapping Φ : W → A^{n′} ⊕ R^{m′} is of the following form:

X^{i′} = φ^{i′} + ∑_{|p|=1}^{q} \frac{1}{p!} \frac{D^{p}φ^{i′}}{Dx^{p}} X^{∘}^{p}, y^{u′} = φ^{u′}(x^{i}, y^{u}),  (29)

where the functions φ^{i′} are of the form

φ^{i′} = φ^{i′0}(x^{i}) + φ^{i′\overline{a}}(x^{i})e\overline{a} + φ^{i′â}(x^{i}, y^{u})eâ.

Proof. To derive the formula (29) we need to integrate the corresponding system of differential equations (3). Since the equations (3) are the conditions for the tangent mapping d_{X}Φ to be A-linear, let us first establish the structure of an A-linear mapping

ψ : A^{n} ⊕ R^{m} → A^{n′} ⊕ R^{m′}.  (30)

The mapping ψ can be represented as the sum of linear mappings ψ_{1} : A^{n} → A^{n′}, ψ_{2} : R^{m} → A^{n′}, ψ_{3} : A^{n} → R^{m′}, and ψ_{4} : R^{m} → R^{m′} such that

U ⊕ u = ψ(V ⊕ v) = (ψ_{1}(V) + ψ_{2}(v)) ⊕ (ψ_{3}(V) + ψ_{4}(v)).

From the relation (27), which defines the action of A on the A-module A^{n} ⊕

R^{m}, it follows that ψ is A-linear if and only if for any α ∈ A^{∘} there holds

(ψ_{1}(αV) ⊕ ψ_{3}(αV)) = (αψ_{1}(V) + αψ_{2}(v)) ⊕ 0.  (31)

Since any element of A^{n}^{∘} is a linear combination (with real coefficient) of el-

ements of the form αV, where α ∈ A^{∘}, from (31) we obtain the following necessary and sufficient conditions for ψ to be A-linear:

1) ker ψ_{3} = A^{n}^{∘}. Since A^{n} = R^{n} ⊕ A^{n}^{∘}, this condition is equivalent to the fact that ψ_{3} is of the form ψ_{3} = ψ′_{3} ∘ π_{0}^{q}, where ψ′_{3} : R^{n} → R^{n′}.

2) im ψ_{2} ⊂ (Ann(A^{∘}))^{n}, where Ann(A^{∘}) is the annihilator of A^{∘}.

3) ψ_{1} is an A-linear mapping.

Applying the conditions 1) — 3) to d_{X}Φ, we obtain the following necessary and sufficient conditions for a mapping (28) to be A-smooth:

1) ∂φ^{u′}/∂x^{i\overline{a}} = 0, ∂φ^{u′}/∂x^{iâ} = 0.

2) ∂φ^{i′\overline{a}}/∂y^{u} = 0.

3) {X^{i}} ↦ {Φ^{i′}(X^{i}, y_{0}^{u})} is an A-smooth mapping from W ∩ (A^{n} ⊕ {y_{0}^{u}}) to A^{n}.

Then the formula (29) follows from Theorem 1. □

4. Examples of smooth manifolds over algebras

In this section we give examples of manifolds modelled on modules over algebras and functors on categories of manifolds over algebras.

By analogy with the corresponding examples of complex manifolds (see, for example, [15], §2.2), for any module L of the category A−Mod one can construct the following two examples of L-manifolds:

1. Let dim_{R}L = k, and G ⊂ L be the additive subgroup in L generated by a collection {u_{1}, u_{2}, …, u_{m}} of linear independent over R elements of L, where m ≤ k. Then L/G is an L-manifold diffeomorphic to the product (S^{1})^{m} × R^{k−m}.

2. Let g : L \ {0} → L \ {0} be a mapping which maps u ∈ L to λu, where λ is a real number greater than 1, and G an infinite cyclic group generated by the diffeomorphism g. Then L \ {0}/G is an L-manifold diffeomorphic to the product S^{1} × S^{k−1}. This manifold belongs to the class of Hopf manifolds (see also Example 7 below).

3. Weil bundles. For any local Weil algebra (see §1) A and smooth real manifold M_{n}, the Weil bundle T^{A}M_{n} of infinitely near points of A-type (briefly, A-points) over M_{n} [50] is an A-smooth manifold modelled on the A-module A^{n}. This fact was discovered by A.P.Shirokov in [36].

The Weil bundle T^{A}M_{n} is defined as follows. An A-point near to x ∈ M_{n} is a homomorphism x′ : C^{∞}(M_{n}) → A such that the real part of x′(f) ∈ A coincides with f(x). The set T^{A}M_{n} of all A-points near to points of M_{n} can be endowed with a structure of smooth manifold. The correspondence which assigns to a real manifold M_{n} the Weil bundle T^{A}M_{n} and to a smooth mapping φ : M_{n} → W_{k} the mapping φ^{A} : M_{n}^{A} ∋ x′ ↦ x′ ∘ φ^{∗}, where φ^{∗} : C^{∞}(W_{k}) ∋ g ↦ g ∘ φ ∈ C^{∞}(M_{n}), is a functor called the Weil functor. The geometry of Weil bungles was studied by P.C. Yuen [55], L.-N. Patterson [29], A. Morimoto [26], A.Ya. Sultanov [42] and other researchers.

The bundle T^{A}M_{n} is naturally equivalent to the bundle of A-velocities (Ajets) studied in [38], [40], [17]. Note that Weil functors appear also as product preserving functors (see [17]), which are studied in many papers (see [17], [23], [8]).

The bundle T_{N}^{q}M_{n} of (N, q)-velocities introduced by C.Ehresmann [10] is isomorphic to the Weil bundle T^{R(N,q)}M_{n} of R(n, q)-velocities and carries a structure of R(n, q)-smooth manifold. The q-frame bundle B^{q}M_{n} ([10], [54]) is an open submanifold in T_{n}^{q}M_{n} and, therefore, carries a structure of R(n, q)- smooth manifold as well.

4. Transverse Weil bundles. Similarly to the preceding example, one can define the transverse Weil functor on the category of foliated manifolds. Let (M_{n}, F) be a foliated manifold and C_{b}^{∞}(M_{n}, x) the algebra of germs of basic functions on M_{n} at x. A transverse A-point near to x ∈ M_{n} is a homomorphism x′ : C_{b}^{∞}(M_{n}, x) → A such that the real part of x′(f) ∈ A coincides with f(x). The set T_{tr}^{A}M_{n} of all transverse A-points near to points of M_{n} can be endowed with a structure of smooth fiber bundle over M_{n}. The bundle T_{tr}^{A}M_{n} is naturally equivalent to the bundle of transverse A-velocities (A-jets) and carries a structure of A-smooth manifold modelled on the module A^{k} ⊕ R^{m} (m is the dimension of leaves of F, k + m = n) [40]. Transverse Weil functors and transverse bundles of higher order were studied by Z.Pogoda [31], and R.Wolak [52].

The transverse q-frame bundle B_{tr}^{q}M_{n}, as an open submanifold in T_{tr}^{R(q,k)}M_{n}, carries a structure of R(k, q)-smooth manifold.

In the more general situation of a tower of submersions, V.V. Vishnevsky introduced so called semitangent bundles of higher order, which also carry structures of smooth manifolds over local algebras [46].

5. Lie groups over local algebras and homogeneous spaces. Let G^{A} be a Lie group over an algebra A, that is, G^{A} is a Lie group and an A-smooth manifold and the composition in G^{A} is an A-smooth mapping. Let H be a discrete subgroup in G^{A}, then G^{A}/H is an A-smooth manifold.

For example, let ℜ be a discrete subring in A (if A admits a basis containing the unit such that all structure constant are integers, one can take the subring Z_{A} ⊂ A consisting of elements whose coordinates are integers). Then in the Lie group G^{A} of triangular matrices of the form

( \begin{matrix}1 & α_{1}^{1} & … & α_{n}^{1} \\ 0 & 1 & … & α_{n}^{2} \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & … & 1 \end{matrix} )

the set of matrices whose entries belong to ℜ is a discrete subgroup.

Weil bundles over real Lie groups are examples of Lie groups over algebras. Let G be a real Lie group with composition ∘ : G × G → G and 𝔤 the Lie algebra of G. The A-prolongation of the mapping ∘ is an A-smooth mapping ∘^{A} : J^{A}G × J^{A}G → J^{A}G, and the bundle J^{A}G with composition ∘^{A} is a Lie group over A. The Lie algebra of J^{A}G is isomorphic to A ⊗ 𝔤.

In particular, the tangent bundle T G carries a structure of Lie group over the algebra R(ε) of dual numbers.

Examples of Lie groups over algebras which are not prolongations of real Lie groups can be constructed as follows.

Consider the following set G^{A} of triangular matrices over A:

( \begin{matrix}1 & εX^{1} & X^{1} \\ 0 & 1 & X^{2} \\ 0 & 0 & 1 \end{matrix} )  (32)

where ε ∈ A is an element satisfying the relation ε^{2} = 0. G^{A} is a two-di-mensional Lie group over A. Since the bracket in the Lie algebra 𝔤^{A} of G^{A} is defined by the relation [E_{1}, E_{2}] = εE_{1} and the quotient group G of G^{A} by

the normal subgroup corresponding to the ideal A^{∘} ⊗ 𝔤 is commutative, G^{A} cannot be a prolongation of a real Lie group. The canonical epimorphism p : G^{A} → R^{2} assigns to a matrix (32) the real matrix whose entries are real parts of the corresponding entries of (32).

Let A = R(ε) and H be the discrete subgroup consisting of matrices (32) whose entries are whole dual numbers (that is, numbers a + bε with a, b ∈ Z). Then the homogeneous space G^{R(ε)}/H is a smooth manifold over R(ε).

Similarly, taking an element ε ∈ A such that ε^{3} = 0, one can construct a three-dimensional Lie group G^{A} over A consisting of matrices of the following form:

( \begin{matrix}1 & εX^{1} & ε^{2}X^{1} & X^{1} \\ 0 & 1 & ε^{2}X^{2} & X^{2} \\ 0 & 0 & 1 & X^{3} \\ 0 & 0 & 0 & 1 \end{matrix} )  (33)

This Lie group also cannot be isomorphic to the A-prolongation of a three-di-mensional real Lie group.

If A = R(ε^{2}), taking in the Lie group G^{R(ε^{2})} of matrices of the form (33) the subgroup H consisting of matrices whose entries are whole elements of the algebra A = R(ε^{2}) (a + bε + cε^{2} with a, b, c ∈ Z) we obtain the homogeneous space G^{R(ε^{2})}/H, which is a smooth manifold over R(ε^{2}).

6. Quotient manifolds of Weil bundles. A-smooth manifolds can be obtained as quotient manifolds with respect to properly discontinuous free actions [14] of groups of A-diffeomorfisms on Weil bundles.

1) Let S^{2n+1} → CP^{n} be the principal Hopf bundle with structure group S^{1} (see [24]). The group Z of whole numbers acts on S^{2n+1} by rotations of fibers,

Z ∋ a ↦ (φ_{a} : S^{2n+1} → S^{2n+1}),

where φ_{a} rotates each fiber through the angle 2πaα. Let s : S^{2n+1} → J^{A}S^{2n+1} be a nonvanishing section. Then the action

Z ∋ a ↦ (s^{A} ∘ φ_{1}^{A})^{a} : J^{A}S^{2n+1} → J^{A}S^{2n+1}

is free and properly discontinuous. Thus, the quotient manifold of J^{A}S^{2n+1} by this action is an A-smooth manifold.

2)Let G_{r} be an r-dimensional Lie group and p : P → M_{n} a principal bundle with structure group G_{r}. The fundamental vector fields {E_{i}}_{i=1...r} on P which correspond to the vectors of a basis of the Lie algebra 𝔤 of G_{r} are sections of the bundle π_{0}^{1} : T P → P and generate a properly discontinuous free action of the group Z^{r} on T P. The quotient manifold is a smooth manifold over R(ε).

7. Hopf manifolds. (see [11] and [15] for the cases of real and complex affine manifolds). Let L_{1}, …, L_{n} be a family of finite-dimensional A-modules and α_{1}, …, α_{n} a collection of elements of A such that |α_{i}^{0}| > 1, where α^{0} is the real part of α. Consider the infinite cyclic group of diffeomorphisms of the A-module L = L_{1} ⊕ ⋯ ⊕ L_{n} generated by the isomorphism (of A-modules)

φ : L ∋ x_{1} ⊕ ⋯ ⊕ x_{n} ↦ α_{1}x_{1} ⊕ ⋯ ⊕ α_{n}x_{n} ∈ L .

The quotient L \ {0}/G is an A-smooth manifold modelled on L.

8. An analogue of the Weil functor for A = \overline{A} ⊕ I.

Suppose that a local algebra A is a semidirect sum A = \overline{A}⊕I of a subalgebra \overline{A} ≅ A/I and an ideal I. In this case there is a functor of A-prolongation from the category of finite-dimensional \overline{A}-smooth manifolds to the category of finite-dimensional A-smooth manifolds.

Let M\overline{A} ; n be an A-smooth manifold with maximal atlas \{(\overline{U}_{α}, \overline{h}_{α} : \overline{U}_{α} → \overline{U} ′_{α})\}_{α∈A}. Consider the disjoint union of open subsets p^{−1}(\overline{U} ′_{α}) = \overline{U} ′_{α} × I^{n} ⊂ A^{n} over all α ∈ A and define the equivalence relation on this union by setting X ∈ p^{−1}(\overline{U} ′_{α}) and Y ∈ p^{−1}(\overline{U} ′_{β}) to be equivalent if Y = (\overline{h}_{β} ∘ \overline{h} _{α}^{−1})^{A}(X). Let

E^{I}(M\overline{A} ; n) = ⨆_{α∈A} p^{−1}(\overline{U} ′_{α})/ ∼

be the quotient set and g_{α} : p^{−1}(\overline{U} ′_{α}) → E^{I}(M\overline{A} ; n) the restriction of the canonical projection. The atlas (g_{α}(\overline{U} ′_{α} ×I^{n}), h_{α}), where h_{α} : g_{α}(\overline{U} ′_{α} ×I^{n}) → \overline{U} ′_{α} ×I^{n} is the inverse mapping to g_{α}, endowes E^{I}(M\overline{A} ; n) with a structure of n-dimension-al A-smooth manifold. The projection p : A^{n} → \overline{A}^{n} induces a locally trivial fiber bundle p : E^{I}(M\overline{A} ; n) → M\overline{A} ; n with standard fiber I^{n}. Since (\overline{h}_{β} ∘ \overline{h} _{α}^{−1})^{A} sends elements belonging to \overline{A}^{n} ⊂ A^{n} to elements of \overline{A}^{n} ⊂ A^{n}, then the set of points with zero fiber coordinates is a well-defined submanifold in E^{I}(M\overline{A} ; n). We shall call this submanifold the zero section of E^{I}(M\overline{A} ; n). The inclusion i : \overline{A} → A gives rise to the corresponding monomorphism of A-modules i : \overline{A}^{n} → A^{n} and embedding i : M\overline{A} ; n → E^{I}(M\overline{A} ; n) of M\overline{A} ; n onto the zero section in E^{I}(M\overline{A} ; n). This embedding is a morphism in the category A−Man. In order to assign to an \overline{A}-smooth mapping \overline{Φ} : M\overline{A} ; n → W \overline{A} ; k an A-smooth mapping E^{I}(\overline{Φ}) : E^{I}(M\overline{A} ; n) → E^{I}(W \overline{A} ; k), it suffices to prolong the coordinate expressions (23) of the mapping \overline{Φ} to the corresponding expressions (24).

Any local algebra A is the semidirect sum A = R ⊕ A^{∘}. In this case,

the functor of A-prolongation E^{A^{∘}}, which assigns to a real manifold M_{n} the

A-smooth manifold E^{A^{∘}}M_{n}, is naturally equivalent to the Weil functor T^{A}. Another example of the situation when a local algebra is a semidirect sum of a subalgebra and an ideal is the representation of the tensor product A′ ⊗ A of two local algebras in the form A′ ⊗ A = A′ ⊕ A′⊗A^{∘}. This being the case,

the functor of A′ ⊗ A-prolongation E^{A′⊗A^{∘}} is naturally equivalent to the Weil functor T^{A}.

9. Integrable almost tangent and almost transverse structures. For the algebra R(ε) of dual numbers, the notion of a structure of R(ε)-

smooth manifold is equivalent to that of an integrable almost tangent structure. Integrable almost tangent structures were studied by many researchers (see [3], [4], [5], [7], [20] [44], [21]). As was proved by M.A.Malakhaltsev [21], for any orientable foliation F on a two-dimensional torus S^{1} × S^{1} there exists a structure of one-dimensional manifold over R(ε) with respect to which F is the canonical foliation.

Manifolds with integrable almost transverse structures studied by T. van Duc [9] belong to the class of R(ε)-smooth manifolds modelled on R(ε)- modules of the form R(ε)^{n} ⊕ R^{m} (see [40]).

5. Homotopy groupoids of an n-dimensional manifold over a local algebra

With a manifold M_{n}^{A} over a local algebra A modelled on the A-module A^{n} one can naturally associate certain bundles and groupoids which play important role in the study of geometry and topology of M_{n}^{A}. On these bundles and groupoids there arise natural structures of manifolds of the category A−Man.

The set of all A-diffeomorphisms between domains of the A-module A^{n} is a pseudogroup, we shall denote it by Γ(A^{n}). By Theorem 1, a diffeomorphism (Φ : U → V) ∈ Γ(A^{n}) is given by equations of the form (11)

Y^{i} = φ^{i} + ∑_{|p|=1}^{q} \frac{1}{p!} \frac{D^{p}φ^{i}}{Dx^{p}} X^{p}^{∘},  (34)

where φ^{i} : U ⊂ A^{n} → A are basic with respect to the canonical A^{n}^{∘}-fo-liation smooth mappings. A structure of A-smooth manifold modelled on the A-module A^{n} on a real manifold M is given by an Γ(A^{n})-atlas, i. e., an atlas {h_{α} : U_{α} ⊂ M_{n}^{A} → U′_{α} ⊂ A^{n}}_{α∈A} whose charts take values in A^{n} and coordinate transformations h_{α} ∘ h_{β}^{−1} belong to the pseudogroup Γ(A^{n}). Γ(A^{n})-manifold will be denoted by M_{n}^{A} and called an n-dimensional A-smooth manifold.

Each ideal I of A generates the invariant submodule (see §2) I^{n} ⊂ A^{n}, which gives rise to the corresponding foliation F^{I} on M_{n}^{A} called the canonical I^{n}-fo-

liation. The canonical A^{n}^{∘}-foliation on M_{n}^{A} which corresponds to the maximal

ideal A^{∘} will be denoted by F^{∘}.

Restrictions of a diffeomorphism Φ ∈ Γ(A^{n}) to leaves of the canonical A^{n}^{∘}- foliation on A^{n} are given by equations of the form

Y^{∘} ^{i} = ∑_{|p|=0}^{q} φ_{p}^{i}X^{∘}^{p}, φ_{p}^{i} ∈ A, φ_{0}^{i} = Y^{∘} _{0}^{i} ∈A^{∘} .  (35)

The set of all diffeomorphisms Φ :A^{n}^{∘}→A^{n}^{∘} of the form (35) is a Lie group, which is denoted by D_{n}(A) and called the A-affine differential group [37],

[38]. Thus, each leaf of the canonical A^{n}^{∘}-foliation on M_{n}^{A} carries a structure of (X, G)-manifold in the sense of W.Thurston [43] (see also [1]) for X =A^{n}^{∘}

and G = D_{n}(A), and M_{n}^{A} carries a structure of tangential (A^{n}^{∘}, D_{n}(A))-fo-

liation. (A^{n}^{∘}, D_{n}(A))-manifolds are also called A^{n}^{∘}-manifolds. In the general form, tangential (X, G)-foliations were studied by M.A. Malakhaltsev [22].

Studying the canonical I^{n}-foliation on M_{n}^{A}, it is convenient to choose a basis {e_{a}} (9) in A of special form so that a part of this basis form a basis of the ideal I. One can obtain such a basis using a Jordan-Hölder composition series (8) containing the ideal I = I_{k}:

A ⊃A^{∘}= I_{1} ⊃ I_{2} ⊃ … ⊃ I_{k} ⊃ … ⊃ I_{m} ⊃ 0,

and taking e_{a} ∈ I_{a}, e_{a} ∉ I_{a+1}. Then the elements e_{a} for a ≥ k form a basis in I.

Fixing in (35) x^{ja} = 0 for a < k and assuming that φ_{0}^{i} ∈ I we obtain a polynomial diffeomorphism

Ψ = Φ|I^{n} : I^{n} → I^{n}

of the form

Y̌^{i} = ∑_{|p|=0}^{q} ψ_{p}^{i}X̌^{p},  (36)

where

ψ_{p}^{i} ∈ A, ψ_{0}^{i} = Y̌ _{0}^{i} ∈ I, det(ψ_{j}^{i}) ∉A^{∘} .

One can consider that in (36) ψ_{p}^{i} ∈ A/Ann(I^{|p|}), where (I)^{|p|} denotes the |p|-th power of the ideal I. The set of diffeomorphisms of the form (36) is a Lie group which acts transitively on I^{n}. Denote this group by D_{n}(I). From (34) it follows that each leaf of the canonical I^{n}-foliation F^{I} on M_{n}^{A} carries a structure of (I^{n}, D_{n}(I))-manifold. For breavity, we shall refer to (I^{n}, D_{n}(I))-manifolds as I^{n}-manifolds.

I^{n}-manifolds for n = 0, 1, 2, … form a category whose morphisms are mapping locally given by equations of the form (36):

Y̌^{i} = ∑_{|p|=0}^{q} ψ_{p}^{i}X̌^{p}.

Suppose, X is a real analytic manifold and a Lie group G acts on X by real analytic diffeomorphisms. Then extensions of an X-chart h : M ⊃ U → U′ ⊂ X along paths on (X, G)-manifold M determine a mapping D : M̃ → X from the universal covering space M̃ of M to X called the developing map of M [43], [1]. An (X, G)-manifold M is said to be complete if the developing map D : M̃ → X is a covering [43], [1].

The canonical I^{n}-foliation on M_{n}^{A} is said to be complete if its leaves are all

complete I^{n}-manifolds. A manifold M_{n}^{A} is said to be complete if its A^{n}^{∘}-foliation is complete.

Doubling of an ideal I of a local algebra A. Let I be an ideal of a local algebra A, \overline{A} = A/I the quotient algebra, p : A → \overline{A} the canonical ephimorphism, p ⊕ p the direct sum of two copies of p and Δ : \overline{A} ∋ \overline{X} ↦ (\overline{X}, \overline{X}) ∈ \overline{A} ⊕ \overline{A} the diagonal embedding. We define the local algebra Â_{I} to be the inverse image of the diagonal Δ(\overline{A}) ⊂ \overline{A} ⊕ \overline{A} with respect to the ephimorphism p ⊕ p. The image Δ(A) of A under the diagonal embedding Δ : A → A ⊕ A is contained in Â_{I}. The corresponding embedding of A in Â_{I} will be denoted by Δ̂ : A → Â_{I}. We have the following commutative diagram with exact rows

0 ⟶ I ⊕ I ⟶^{i} A ⊕ A ⟶^{p⊕p} \overline{A} ⊕ \overline{A} ⟶ 0
↑id  ↑i  ↑Δ
0 ⟶ I ⊕ I ⟶^{i}  Â_{I}  ⟶^{p}  \overline{A} ⟶ 0
↑Δ  ↑Δ̂  ↑id
0 ⟶ I ⟶^{i} A ⟶^{p}  \overline{A} ⟶ 0

(37)

where the symbols i denote inclusions and p projections.

Denote by A_{loc} the category of local algebras and by A_{loc}^{2} the category of pairs consisting of a local algebra and its ideal (a morphism from (A_{1}, I_{1}) to (A_{2}, I_{2}) in the category A_{loc}^{2} is a unital homomorphism φ : A_{1} → A_{2} such that φ(I_{1}) ⊂ I_{2}). The correspondence which assigns to a pair (A, I) the algebra Â_{I} is a functor from A_{loc}^{2} to A_{loc}.

Let p_{k} (k = 1, 2) be the projection of A ⊕ A onto k-th direct summond and i_{k} the embedding of I into I ⊕ I as k-th direct summond. For breavity, the composition p_{k} ∘ i : Â_{I} → A of the mapping p_{k} and the embedding i from the middle column of the diagram (37) will be denoted by the same symbol p_{k}. Similarly, the composition i ∘ i_{k} : I → Â_{I}, where i is the embedding from the middle row of the diagram (37), will be denoted by i_{k}. Let \overline{k} = 1 if k = 2, and \overline{k} = 2 if k = 1. The following sequences are exact:

0 ⟶ I \underrightarrow{i_{k}} Â_{I} \underrightarrow{p \overline{k}} A ⟶ 0 .  (38)

The same symbols p_{k} and i_{k} will be used for designation of the corresponding mappings of modules p_{k} : Â_{I}^{n} → A^{n} and i_{k} : I^{n} → Â_{I}^{n}.

An arbitrary A-smooth function F : U ⊂ A^{n} → A is of the form (see (11))

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

where f : U → A is a basic (projectable) function with respect to the canonical

A^{n}^{∘}-foliation on U. The embedding Δ̂ : A → Â_{I} allows to consider A as a subalgebra in Â_{I}. An A-smooth function F can be uniquely prolonged to an Â_{I}-smooth function F̂ : p_{1}^{−1}(U) ⊂ Â_{I}^{n} → Â_{I} such that F̂|U = F. This function F̂ is given as follows:

Y = f̂+ ∑_{|p|=1}^{q} \frac{1}{p!} \frac{D^{p}f̂}{Dx^{p}}X̂^{∘}^{p},  (39)

where f̂ = Δ̂ ∘ f ∘ p_{1} : p_{1}^{−1}(U) → Â_{I}, {X̂^{∘}^{i}} ∈ p_{1}^{−1}(U) ⊂A^{n}^{∘}.

In case a function F is given by a complicated expression its prolongation F̂ will be denoted as follows: F .

The canonical osculating bundles. Consider an A-smooth manifold M_{n}^{A} with an A^{n}-atlas {h_{α} : U_{α} ⊂ M_{n}^{A} → U′_{α} ⊂ A^{n}}_{α∈A} and coordinate transformations h_{α} ∘ h_{β}^{−1} of the form (34). With the canonical I^{n}-foliation on M_{n}^{A} one can naturally associate the bundle π_{I} : O_{I}^{V} (M_{n}^{A}) → M_{n}^{A} with standard fiber I^{n}, structure group D_{n}(I), and atlas

{H_{α} : π_{I}^{−1}(U_{α}) → U′_{α} × I^{n}}_{α∈A}  (40)

whose transition functions are of the form H_{α} ∘ H_{β}^{−1} = (h_{α} ∘ h_{β}^{−1}) . By construction, the manifold O_{I}^{V} (M_{n}^{A}) carries a structure of n-dimensional Â_{I}- smooth manifold. In addition, the pair (p_{1} : Â_{I} → A, π_{I} : O_{I}^{V} (M_{n}^{A}) → M_{n}^{A}) is a morphism in the category A−Man. The bundle π_{I} : O_{I}^{V} (M_{n}^{A}) → M_{n}^{A} will be called the canonical osculating I^{n}-bundle of M_{n}^{A}. The canonical osculating

A^{n}^{∘}-bundle of M_{n}^{A} will be denoted as follows: π : O^{V} (M_{n}^{A}) → M_{n}^{A}.

Every n-dimensional Â_{I}-smooth manifold carries two canonical I^{n}-foliations F_{1}^{I} and F_{2}^{I} corresponding, respectively, to the embeddings i_{1} and i_{2} of I into Â_{I}. In terms of local Â_{I}-coordinates, {X̂^{i}} the foliations F_{k}^{I} are given, respectively, by the equations p_{k}(X̂^{i}) = const. The canonical I^{n}-foliation F_{2}^{I} of the bundle O_{I}^{V} (M_{n}^{A}) is formed by the fibers of this bundle. The foliation F_{1}^{I} on O_{I}^{V} (M_{n}^{A}) can be described as follows. The bundle O_{I}^{V} (M_{n}^{A}) is a foliated bundle [25],

[12] with respect to the canonical I^{n}-foliation F^{I} on M_{n}^{A}, and the foliation F_{1}^{I} is the lifted foliation with respect to F^{I}. Under the projection π_{I}, a leaf of the lifted foliation F_{1}^{I} covers the corresponding leaf of the foliation F^{I}. The foliation F_{1}^{I} determines a flat partial [12] connection Γ_{I}^{V} in the bundle O_{I}^{V} (M_{n}^{A}) (a connection along leaves of the foliation F^{I}) whose horizontal distribution is given by the equations dX_{2}^{i} = 0. The parallel translation with respect to the connection Γ_{I}^{V} along a leaf curve γ on M_{n}^{A} gives covering curves γ̃ lying in leaves of the lifted foliation.

Homotopy groupoids of M_{n}^{A}. The fundamental groupoid Π(W) of a topological space W is defined as the set of homotopy classes of paths in W with fixed ends. There are natural projections π_{0} : Π(W) → W and π_{1} : Π(W) → W which assign to the class [γ] of a path γ : [0, 1] → W its initial and terminal points γ(0) and γ(1). The homotopy groupoid Π(F) of a foliation F on a smooth manifold W is defined as the fundamental groupoid of W endowed with the leaf topology (i. e., Π(F) is the set of homotopy classes of leaf paths) [32]. The homotopy groupoid Π(F) is endowed with the quotient topology determined by the topologies on [0, 1] and W and the compact-open topology on the space of paths in W. On Π(F) there arises also a structure of smooth manifold. The homotopy groupoid of the canonical I^{n}-foliation F^{I} on an A-smooth manifold M_{n}^{A} will be denoted by Π_{I}(M_{n}^{A}).

Let p_{k} : Â_{I} → A be the restriction of the projection of the direct sum onto its k-th summond (see (38)) and ι : M_{n}^{A} ∋ X ↦ [X] ∈ Π_{I}(M_{n}^{A}) the embedding which assigns to X ∈ M_{n}^{A} the homotopy class of the constant path X : [0, 1] ↦ X ∈ M_{n}^{A}.

Theorem 4. The homotopy groupoid Π_{I}M_{n}^{A} of the canonical I^{n}-foliation F^{I} on M_{n}^{A} carries a structure of n-dimensional Â_{I}-smooth manifold with respect to which the pairs (p_{k}, π_{k}), k = 1, 2, and (Δ̂, ι) are morphisms in the category A−Man.

Proof. Consider a leaf L_{X_{0}}^{I} of the foliation F^{I} through X_{0} ∈ M_{n}^{A}, an A^{n}- chart (U, h : U → U′ ⊂ A^{n}) on M_{n}^{A} such that U ∋ X_{0} and U′ is an open coordinate parallelepiped in A^{n} (therefore, U′ is a simple open set for the canonical foliations on A^{n}), and a continuous path γ : [0, 1] → L_{X_{0}}^{I} joining X_{0} and X_{1} ∈ L_{X_{0}}^{I}. Let (π_{I}^{−1}(U), H) be the Â_{I}-chart on the bundle O_{I}^{V} (M_{n}^{A}) induced by (U, h), and H_{X_{0}} = pr_{2} ∘ H|π_{I}^{−1}(X_{0}) : π_{I}^{−1}(X_{0}) → I^{n}, where pr_{2} is the projection of U′ × I^{n} onto I^{n} (see (40)). The chart (U, h) also induces the (I^{n}, D_{n}(I))-chart (U ∩ L_{X_{0}}^{I}, h_{X_{0}}^{I}) on the leaf L_{X_{0}}^{I}. Since coordinate changes on the leaf L_{X_{0}}^{I} are polynomial transformations, the mapping h_{X_{0}}^{I} can be extended along the path γ (see, e. g., [1]). Assigning to the class [γ] ∈ Π_{I}(M_{n}^{A}) the point D_{I}([γ]) = H_{X_{0}}^{−1}(h_{X_{0}}^{I}(X_{1})) ∈ O_{I}^{V} M_{n}^{A}, we obtain a mapping

D_{I} : Π_{I}(M_{n}^{A}) → O_{I}^{V} (M_{n}^{A}).  (41)

The mapping D_{I} is a local homeomorphism, which allows to introduce on Π_{I}(M_{n}^{A}) a structure of Â_{I}-smooth manifold with respect to which the mapping D_{I} is a local Â_{I}-diffeomorphism. It is easily verified by direct calculations in local coordinates that the projections π_{k} and embedding ι are morphisms in A−Man. □

In the general case, the topology of the homotopy groupoid Π(F) of a foliation F on a manifold W is not Hausdorff. As was proved by P. Dazord and G. Hector [6], Π(F) is Hausdorff if and only if the foliation F has no vanishing cycles. Since I^{n} is obviously simply connected, a complete simply connected I^{n}-manifold M_{n}^{I} is isomorphic to I^{n}. Therefore, in the case the canonical I^{n}-fo-liation is complete, the mapping (41) is an Â_{I}-diffeomorphism. Consequently, the homotopy groupoid Π_{I}M_{n}^{A} is a Hausdorff manifold. Thus, there holds the following

Proposition 3. If the canonical I^{n}-foliation F^{I} of an A-smooth manifold M_{n}^{A} is complete, it has no vanishing cycles.

In particular, the canonical I^{n}-foliation F^{I} on a complete A-smooth manifold M_{n}^{A} has no vanishing cycles. □

Denote by L̃_{X_{0}}^{I} = π_{0}^{−1}(X_{0}) ⊂ Π_{I}(M_{n}^{A}) the set of homotopy classes of leaf paths (for the canonical I^{n}-foliation) on M_{n}^{A} with initial point X_{0}. L̃_{X_{0}}^{I} is the universal covering space of the leaf L_{X_{0}}^{I}. The mapping H ∘D_{I} restricted to L̃_{X_{0}}^{I} coincides with the developing map of the I^{n}-manifold L_{X_{0}}^{I}. For this reason, we shall call the mapping D_{I} the developing map of the canonical I^{n}-foliation F^{I} on M_{n}^{A}. The mapping D^{∘} = D_{A^{∘}} will be called the developing map of M_{n}^{A}.

6. Holonomy groupoids of an n-dimensional manifold over a local algebra

With each leaf L of a foliation F on a manifold W one can associate the holonomy representation [25], [32], which is the homomorphism from the fundamental group Π_{1}(L) of L to the group of germs of diffeomorphisms of a local transversal determined by sliding of local transversals along leaf paths. With a leaf L^{I} of the canonical I^{n}-foliation of an A-smooth manifold M_{n}^{A}, one can also associate the holonomy representation of L^{I} viewed as an (I^{n}, D_{n}(I))-manifold [43], [1]. In this section we consider a holonomy representation of L^{I} which is generated by the A-smooth structure of M_{n}^{A} and determines the two above indicated representations.

Extension of an A-chart along a leaf path [41]. Consider a leaf LA^{∘} ; X_{0} of

the canonical A^{n}^{∘}-foliation F^{∘} on M_{n}^{A} through X_{0} and an A^{n}-chart (U, h : U → U′ ⊂ A^{n}) such that U ∋ X_{0} and U′ is an open coordinate parallelepiped in A^{n}.

Let γ : [0, 1] → LA^{∘} ; X_{0} be a path joining X_{0} and X_{1} ∈ LA^{∘} ; X_{0}. The mapping h can be extended along γ as follows. Let {(U_{k}, h_{k} : U_{k} → U′_{k})}, k = 1, …, s, be a collection of A-charts on M^{A} satisfying the following conditions: U′_{k} is an open coordinate parallelepiped in A^{n}, (U_{1}, h_{1}) = (U, h), and there is a subdivision t_{0} = 0 < t_{1} < ⋯ < t_{m} = 1 of [0, 1] such that γ([t_{k}, t_{k+1}]) ⊂ U_{k+1}. An Asmooth transformation Φ_{k} = h_{k} ∘ h_{k+1}^{−1} restricted to a simple neighborhood V_{k} of γ(t_{k}) can be extended to an A-diffeomorphism Φ̃_{k} defined on the domain π_{0}^{q}^{−1}(π_{0}^{q}(V_{k})), therefore on a suitable domain of the form (π_{0}^{q})^{−1}(V), where V ⊂ R^{n} is an open neighborhood of π_{0}^{q}(h_{1}(X_{1})) ∈ R^{n}, one can take the composition Φ_{1} ∘ ⋯ ∘ Φ_{s−1}. Then on some neighborhood W of X_{1} there is defined an A^{n}-chart (W, h(γ)), where h(γ) = Φ_{1} ∘⋯∘Φ_{s−1} ∘h_{s}. This A^{n}-chart is called the extension of the A^{n}-chart (U, h) along γ. The germ of the chart (W, h(γ)) at X_{1} depends only on the homotopy class of the path γ in the leaf

LA^{∘} ; X_{0}.

Holonomy representations of M_{n}^{A}. Consider a leaf L_{X_{0}}^{I} of the canonical I^{n}-foliation F^{I} of M_{n}^{A} through X_{0} and an A^{n}-chart (U, h : U → U′ ⊂ A^{n}) such that U ∋ X_{0}, U′ is an open coordinate parallelepiped in A^{n} containing 0, and h(X_{0}) = 0. Let γ : [0, 1] → L_{X_{0}}^{I} be a loop in L_{X_{0}}^{I} attached at X_{0}. Extending the chart (U, h) along γ, we obtain a new A^{n}-chart (W, h_{γ}) whose domain of definition W contains X_{0}. Denote by α_{X_{0}}^{I}(γ) = α_{X_{0} h}^{I}(γ) the germ of the composition h_{γ} ∘ h^{−1} at 0 ∈ A^{n}. This germ is defined along the entire

submodule A^{n}^{∘}⊂ A^{n} and depends only on the homotopy class of γ. Let now γ_{1} and γ_{2} be two loops in the leaf L_{X_{0}}^{I} attached at X_{0} and γ_{3} = γ_{2} ∗ γ_{1}. Then α_{X_{0}}^{I}(γ_{3}) = α_{X_{0}}^{I}(γ_{2}) ∘ α_{X_{0}}^{I}(γ_{1}). As a result, we obtain a homomorphism

α_{X_{0}}^{I} : Π_{1}(L_{X_{0}}^{I}, X_{0}) ∋ [γ] ↦ α_{X_{0}}^{I}(γ) ∈ Diff_{A}(A^{n}, I^{n}),  (42)

where Diff_{A}(A^{n}, I^{n}) is the group of germs of A-diffeomorphisms of the form p^{−1}(V) ⊂ A^{n} → p^{−1}(V′) ⊂ A^{n}, p : A^{n} → \overline{A}^{n} the canonical projection, and V and V′ are neighborhoods of 0 in \overline{A}^{n}.

The homomorphism α_{X_{0}}^{I} will be called the I-holonomy representation of M_{n}^{A} at X_{0}. Its image α_{X_{0}}^{I}(Π_{1}(L_{X_{0}}^{I}, X_{0})) ⊂ Diff_{A}(A^{n}, I) will be called the I-holonomy group of M_{n}^{A} at X_{0}. The holonomy group is determined up to a change by a conjugate subgroup in Diff_{A}(A^{n}, I^{n}) with respect to an inner automorphism. One can define an I-holonomy representation in terms of the canonical osculating bundle O_{I}^{V} (M_{n}^{A}) which does not depend on the choice of

an A^{n}-chart (U, h), U ⊃ X_{0} (see [41] for the canonical A^{n}^{∘}-foliation).

The homomorphism α^{∘}_{X_{0}} = αA^{∘} ; X_{0} is called the holonomy representation of M_{n}^{A} at X_{0} [41].

A germ of A-diffeomorphism Φ ∈ Diff_{A}(A^{n}, I^{n}) given by equations of the form (34) determines the germ \overline{Φ} ∈ Diff\overline{A}( \overline{A}^{n}, 0) given by the corresponding equations

\overline{Y} ^{i} = ∑_{|p|=1}^{q} \frac{1}{p!} \frac{D^{p}\overline{φ}^{i}}{Dx^{p}} \overline{X}^{∘} ^{p},  (43)

where \overline{φ}^{i} = φ^{i} + I are the residue classes of φ^{i}. Denote by

p^{I} : Diff_{A}(A^{n}, I) → Diff\overline{A}( \overline{A}^{n}, 0)  (44)

the homomorphism which assigns to a germ Φ the germ \overline{Φ}, and by

r^{I} : Diff_{A}(A^{n}, I^{n}) → D_{n}(I)  (45)

the homomorphism which assigns to a germ Φ its restriction Φ|I^{n}. Compairing the definition of the I-holonomy of M_{n}^{A} respectively with the definitions of the holonomy of an (X, G)-manifold [43], [1] and the holonomy of a foliation [25], [32], we obtain the following

Proposition 4. 1) The composition α_{l X_{0}}^{I} = r^{I} ∘ α_{X_{0}}^{I} is the holonomy representation of the leaf L_{X_{0}}^{I} viewed as an (I^{n}, D_{n}(I))-manifold.

2) The composition α_{tr X_{0}}^{I} = p^{I} ∘ α_{X_{0}}^{I} is the holonomy representation of the foliation F^{I} at X_{0}. □

From Proposition 4 it follows that there is a dependence between the holonomy representation α_{tr X_{0}}^{I} of L_{X_{0}}^{I} in the group Diff\overline{A}( \overline{A}^{n}, 0) of germs of \overline{A}-smooth diffeomorphisms of a local transversal to the I^{n}-foliation and the holonomy representation α_{l X_{0}}^{I} of L_{X_{0}}^{I} viewed as an I^{n}-manifold. In the case when A is a semidirect sum A = \overline{A} ⊕ I, one can construct a holonomy representation which is a quotient representation of the representation α_{tr X_{0}}^{I} as well as of the representation α_{l X_{0}}^{I}.

Let now A be a semidirect sum A = \overline{A} ⊕ I. This being the case, the Lie group D_{n}(I) similarly to D_{n}(A) (see Section 5) turns out to be a semidirect product

D_{n}(I) = D̃_{n}(I) × D^{∘}_{n}(I)

of the Lie subgroup D̃_{n}(I) consisting of diffeomorphisms of the form (36) with

ψ_{p}^{i} ∈ \overline{A} and ψ_{0}^{i} = 0, and the normal subgroup D^{∘}_{n}(I) determined by the relations ψ_{j}^{i} = δ_{j}^{i}, ψ_{p}^{i} ∈ I for |p| ≠ 1. Let

τ^{I} : D_{n}(I) → D̃_{n}(I)

be the canonical epimorphism. The composition

α̃_{l X_{0}}^{I} = τ^{I} ∘ α_{l X_{0}}^{I} : Π_{1}(L_{X_{0}}^{I}, X_{0}) → D̃_{n}(I)  (46)

will be called the D̃_{n}(I)-holonomy representation of L_{X_{0}}^{I}.

Proposition 5. Let A be a semidirect sum A = \overline{A}⊕I. Then the D̃_{n}(I)-holo-nomy representation of a leaf L_{X_{0}}^{I} on an A-smooth manifold M_{n}^{A} is determined by the holonomy representation α_{tr X_{0}}^{I}.

Proof. By Theorems 1 and 2, an \overline{A}-smooth mapping \overline{Φ} : \overline{U} ⊂ \overline{A}^{n} → \overline{A}^{k} is of the form (see (23))

\overline{Y} ^{i′} = \overline{φ}^{i′} + ∑_{|p|=1}^{q} \frac{1}{p!} \frac{D^{p}\overline{φ}^{i′}}{Dx^{p}} \overline{X}^{∘} ^{p}, \overline{X}^{i}, \overline{Y} ^{i′} ∈ \overline{A},  (47)

and can be prolonged to the A-smooth mapping Φ̃ = \overline{Φ}^{A} : p^{−1}(\overline{U}) ⊂ A^{n} → A^{k} given by the equations

Y^{i′} = \overline{φ}^{i′}(x^{i}) + ∑_{|p|=1}^{q} \frac{1}{p!} \frac{D^{p}\overline{φ}^{i′}}{Dx^{p}} X^{∘}^{p}, X^{i}, Y^{i′} ∈ A.  (48)

Therefore, to any germ \overline{Φ} ∈ Diff\overline{A}( \overline{A}^{n}, 0) one can associate its A-prolongation \overline{Φ}^{A} ∈ Diff_{A}(A^{n}, I^{n}). As a result, we obtain a monomorphism

i^{I} : Diff\overline{A}( \overline{A}^{n}, 0) → Diff_{A}(A^{n}, I^{n}).

Then from (46) and (48) it follows that

α̃_{l X_{0}}^{I} = r^{I} ∘ i^{I} ∘ α_{tr X_{0}}^{I}.  (49)

□ In the case I = A^{∘}, we get the following expression for the G_{n}^{q}-holonomy

representation of LA^{∘} ; X_{0} (G_{n}^{q} is the group of q-jets of germs of diffeomorphisms (R^{n}, 0) → (R^{n}, 0)) [41]:

α̃A^{∘} ; l X_{0} = j^{q} ∘ αA^{∘} ; tr X_{0} ,

where j^{q} is the mapping which assigns to a germ φ ∈ Diff_{R}(R^{n}, 0) its q-jet.

Holonomy groupoids of M_{n}^{A}. The holonomy groupoid (graph) of a foliation plays a significant role in the study of its geometry and topology (see, e. g., [51], [32], [56]). A similar groupoid can be constructed for an A-smooth manifold M_{n}^{A} on the basis of its I-holonomy representation.

Definition. The I-holonomy groupoid of M_{n}^{A} is the quotient space Γ_{I}(M_{n}^{A}) of the homotopy groupoid Π_{I}(M_{n}^{A}) by the following equivalence relation: [γ_{1}] ∼ [γ_{2}], if γ_{1}(0) = γ_{2}(0), γ_{1}(1) = γ_{2}(1), and α_{γ(0)}^{I}(γ_{2}^{−1} ∗ γ_{1}) = id ∈ Diff_{A}(A^{n}, I^{n}) or, in other words, the germs of the extensions h(γ_{1}) and h(γ_{2}) at γ_{1}(1) of an A^{n}-chart (U, h), U ∋ γ_{1}(0), coincide.

The natural projection p_{I} : Π_{I}(M_{n}^{A}) → Γ_{I}(M_{n}^{A}) is a local homeomorphism and the developing map (41) determines a mapping D̂_{I} : Γ_{I}(M_{n}^{A}) → O_{I}^{V} (M_{n}^{A}) such that the following diagram commutes

Π_{I}(M_{n}^{A})  \underrightarrow{p_{I}}  Γ_{I}(M_{n}^{A})
D_{I}↘  ↙D̂_{I} O_{I}^{V} (M_{n}^{A}) (50)

Let π_{k} : Γ_{I}(M_{n}^{A}) → M_{n}^{A}, k = 0, 1, be the projections similar to those for the homotopy groupoid and ι : M_{n}^{A} ∋ X ↦ [X] ∈ Γ_{I}(M_{n}^{A}) the embedding which assigns to X ∈ M_{n}^{A} the class of the constant path X : [0, 1] ↦ X ∈ M_{n}^{A}. Similarly to Theorem 4 there holds

Theorem 5. The I-holonomy groupoid Γ_{I}(M_{n}^{A}) carries a structure of ndimensional Â_{I}-smooth manifold with respect to which the pairs (p_{k}, π_{k}), k = 1, 2, and (Δ̂, ι) are morphisms in the category A−Man. □

R. Wolak [53] introduced the notion of a holonomy vanishing cycle and proved that the graph of a foliation is Hausdorff if and only if the foliation has no vanishing cycles. A similar statement holds for the I-holonomy groupoid of an A-smooth manifold M_{n}^{A}.

Definition. An I-holonomy vanishing cycle for a manifold M_{n}^{A} is a continuous mapping γ : [0, 1] × [0, 1] → M_{n}^{A} such that for any s ∈ [0, 1] the mapping γ_{s} = γ|[0, 1] × {s} → M_{n}^{A} is a loop in a leaf of the canonical I^{n}-foliation, the loop γ_{0} has nontrivial holonomy: α^{I}(γ_{0}) ≠ id, the holonomy of all the loops γ_{s} for s > 0 is trivial: α^{I}(γ_{s}) = id.

Theorem 6. 1) The I-holonomy groupoid Γ_{I}(M_{n}^{A}) of an A-smooth manifold M_{n}^{A} is Hausdorff if and only if the manifold M_{n}^{A} has no I-holonomy vanishing cycles.

2) An A-smooth manifold M_{n}^{A} with complete I^{n}-foliation has no I-holonomy vanishing cycles.

Proof. The first statement follows from the fact that the elements of Γ_{I}(M_{n}^{A}) which are determined, respectively, by the constant loop X_{0} : [0, 1] ↦ X_{0} and the loop γ_{0} of a vanishing cycle γ with γ_{0}(0) = γ_{0}(1) = X_{0} cannot be separated. Since the manifold O_{I}^{V} (M_{n}^{A}) is Hausdorff (M_{n}^{A} is assumed to be Hausdorff), the second statement follows from the fact that in the case the canonical I^{n}-foliation on M_{n}^{A} is complete all the arrows in the diagram (50) are Â-diffeomorphisms. □

References

[1] Apanasov, B.N., The Geometry of Discrete Groups and Manifolds. Moscow, Nauka, 1991.

[2] Bernstein I.N. and Rosenfeld B.I., Homogeneous spaces of infinite dimensional Lie algebras and characteristic classes of foliations. Russian Math. Surveys (Uspekhi mat.nauk), vol. 28, issue 4 (172), 1973, 103–138.

[3] Brickell F. and Clark R.S., Integrable almost tangent structures. J. Different. Geom., 1974, vol. 9, no. 4, 557–563.

[4] Crampin M., Defining Euler-Lagrange fields in terms of almost tangent structures. Phys. Lett., 1983, vol. 97 A, 466–468.

SMOOTH MAPPINGS OVER WEIL ALGEBRAS 53

[5] Crampin M. and Thompson G., Affine bundles and integrable almost tangent structures. Math. Proc. Cambridge Phil. Soc., 1985, vol. 98, 61–71.

[6] Dazord P. and Hector G., Integration symplectique des variétés de Poisson totalement asphériques, in Symplectic Geometry, Groupoids and Integrable Systems. Séminaire Sud Rhodanien à Berkeley , 1989. Springer Math. SKIP 20, 1991, Springer, 38–72.

[7] De Filippo S., Landi G., Marmo G., and Vilasi G., Tensor fields defining a tangent bundles structure. Ann. Inst. Henri Poincaré, 1989, vol. 50, no. 2, 205–218.

[8] Doupovec M. and Kolář I., On the jets of fibred manifold morphisms. Cahiers de Topol. et Geom. Diff. Categor., vol. XL-1, 1999, 21–30.

[9] Duc Tong van, Structures presque-transverse. J. Different. Geom., 1979, vol. 14, no. 2, 215–219.

[10] Ehresmann C., Les prolongements d’une variété différentiable. I. Calcul des jets, prolongement principal. C. R. Acad. Sci., 1951, t. 233, no. 11, 598–600.

[11] Fried D., Goldman W., and Hirsch M.W., Affine manifolds with nilpotent holonomy. Comm. Math. Helv., 1981, vol. 56, no. 4, 487–523.

[12] Kamber F.W., Tondeur Ph. Foliated bundles and characteristic classes. Lecture Notes in Mathematics, vol. 493, Springer 1975.

[13] Kobayashi S., Manifolds over function algebras and mapping spaces. Tôhoku Math. J., 1989, vol. 41, no. 2, 263–282.

[14] Kobayashi K. and Nomizu K., Foundations of Differential Geometry. Vol. 1, Interscience Publishers, 1963.

[15] Kodaira K., Complex Manifolds and Deformations of Complex Structures. Springer, 1986.

[16] Kolář I., An infinite dimensional motivation in higher order geometry. Proceedings of the Conference on Differential Geometry and Applications. Aug.28 — Sept.1, 1995, Brno, Czech Republic, Brno, 1996, 151–159.

[17] Kolář I., Michor P.W., and Slovák J. Natural Operations in Differential Geometry. Springer, 1993.

[18] Kriegl A. and Michor P.W., Product preserving functors of infinite dimensional manifolds. Arch. Math. 1996, vol. 32, no. 4, 289–306.

[19] Kruchkovich G.I., Hypercomplex structures on manifolds. I. Trudy semin. po vect. i tenz. analis., issue 16, Moscow University, 1972, 174–201.

[20] León M. de, Méndes I., and Salgado M., Integrable p-almost tangent manifolds and tangent bundles of p^{1}-velocities. Acta Math. Hung. 1991, vol. 58, no. 1-2, 45–54.

[21] Malakhaltsev M.A., Structures of manifold over the algebra of dual numbers on the torus. Trudy geometr. semin., issue 22, Kazan University, 1994, 47–62.

[22] Malakhaltsev M.A., (X, G)-foliations. Russian Mathematics (Izv. VUZ), 1996, no. 7, 55–65.

[23] Mikulski W.M., Product preserving bundle functors on fibered manifolds. Archiv. Math., vol 32, 1996, 307–316.

[24] Mishchenko A.S., Vector Bundles and Their Applications. Moscow., Nauka, 1984.

[25] Molino P., Riemannian Foliations. Birkhäuser, 1988.

[26] Morimoto A., Prolongation of connections to bundles of infinitely near points. J. Different. Geom., 1976, vol. 11, no. 4, 479–498.

[27] Papatriantafillou M.H., Partitions of unity on A-manifolds. Internat. J. Math. 1998, vol. 9, no.7, 877–883.

[28] Papatriantafillou M.H., Connections on A-bundles. New Developments in Differential Geometry. Budapest 1996, 307–316, Kluver Acad. Publ., Dordrecht, 1999.

[29] Patterson L.-N., Connexions and prolongations. Canad. J. Math., 1975, vol. 27, no. 4, 766–791.

[30] Pierce R.S., Associative Algebras. Springer., 1982.

[31] Pogoda Z. Horizontal lifts and foliations. Rend. Circ. mat. Palermo. Ser. 2., 1989, vol. 38, suppl. no. 21, 279–289.

[32] Reinhart B.L. Differential Geometry of Foliations. Springer, 1983.

54 VADIM V. SHURYGIN

[33] Scheffers G., Verallgemeinerung der Grundlagen der gewöhnlichen komplexen Funktionen. Berichte Sächs. Akad. Wiss., 1893, Bd. 45, 828–842.

[34] Shirokov A.P., On a type of G-structures determined by algebras. Trudy geometr. semin. VINITI AN SSSR, vol. 1, Moscow, 1966, 425–456.

[35] Shirokov A.P., A note on structures in tangent bundles. Trudy geometr.semin. VINITI AN SSSR, vol. 5, Moscow, 1974, 311–318.

[36] Shirokov A.P., The geometry of tangent bundles and spaces over algebras. Problems of geometry (VINITI), vol. 12. Moscow, 1981, 61–95.

[37] Shurygin V.V., On the theory of differential groups of higher order. Russian Mathematics (Izv.VUZ), 1984, no. 11, 77–81.

[38] Shurygin V.V., Jet bundles as manifolds over algebras. Problems of Geometry (VINITI), vol. 19. Moscow, 1987, 3–22.

[39] Shurygin V.V., Application of the theory of manifolds over algebras to the transversal geometry of foliations. In: To the Memory of Lobachevsky is dedicated. Issue 2. Kazan University, 1992, 119–140.

[40] Shurygin V.V., Manifolds over algebras and their application in the geometry of jet bundles. Russian Math. Surveys (Uspekhi mat.nauk), vol. 48, issue 2 (290), 1993, 75– 106.

[41] Shurygin V.V. Smooth connections and horizontal distributions on manifolds over local algebras. Proceedings of the Conference on Differential Geometry and Applications. Aug. 28 – Sept. 1, 1995. Brno. Czech Republic., Brno, 1996, 309-319.

[42] Sultanov A.Ya., Prolongations of tensor fields and connections to Weil bundles. Russian Mathematics (Izv.VUZ), 1999, no. 9, 81–90.

[43] Thurston W.P., The Geometry and Topology of 3-manifolds. Princeton Univ. Lecture Notes. 1978/1979.

[44] Thompson G. and Schwardmann U., Almost tangent and cotangent structures in the large. Trans. Amer. Math. Soc., 1991, vol. 327, no. 1, 313–327.

[45] Vanžura J., On the geometry and topology of manifolds over algebras. Weiterbildungszentr. Math. Kybern. und Rechentechn. Sect. Math., vol 28, 1978, 133–136.

[46] Vishnevsky V.V., Manifolds over plural numbers and semitangent structures. Problems of Geometry (VINITI), vol. 20. Moscow, 1988, 35–75.

[47] Vladimirov V.S. and Volovich I.V., Superanalysis. I. Differential Calculus. Teoret . i matem. fizika. 1984, vol. 59, no. 1, 3–27.

[48] Wagner V.V., Algebraic theory of differential groups. Dokl. AN SSSR, vol. 80, no. 6, 1951, 845–848.

[49] Wagner V.V., Algebraic theory of tangent spaces of higher orders. Trudy semin. po vect. i tenz. analis., issue 10, Moscow University, 1956, 31–88.

[50] Weil A., Théorie des points proches sur les variététes différentiables. Colloque internat. centre nat. rech. sci., vol 52, Strasbourg, 1953, 111–117.

[51] Winkelnkemper H., The graph of a foliation. Ann. Glob. Anal. Geom., 1983, vol. 1, 51–75.

[52] Wołak R., Normal bundles of foliations of order r. Demonstratio Math., 1985, vol. 18, no. 4, 977–994.

[53] Wolak R.A., Graphs, Ehresmann connections and vanishing cycles. Proceedings of the Conference on differential geometry and applications. Aug. 28 – Sept. 1, 1995, Brno, Czech Republic, Brno, 1996, 345–352.

[54] Yevtushik L.E., Lumiste Ü.G., Ostianu N.M., and Shirokov A.P., Differential-Geomet-ric Structures on Manifolds. Problems of Geometry (VINITI), vol. 9. Moscow, 1979.

[55] Yuen P.C., Sur la notion d’une G-structure déométrique et les A-prolongements de G-structures. C. R. Acad. Sci., 1970, vol. 270, no. 24, A1589–A1592.

[56] Zhukova N.I., The graph of a foliation with Ehresmann connection and stability of leaves. Russian Mathematics (Izv. VUZ), 1994, no. 2, 79–81.

SMOOTH MAPPINGS OVER WEIL ALGEBRAS 55

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