Lobachevskii Journal of Mathematics

http://ljm.ksu.ru

Vol. 18, 2005, 53–105 \begin{matrix}c \\ ◯ \end{matrix} G.N. Bushueva and V.V. Shurygin

G.N. Bushueva and V.V. Shurygin

ON THE HIGHER ORDER GEOMETRY OF WEIL BUNDLES OVER SMOOTH MANIFOLDS AND OVER

PARAMETER-DEPENDENT MANIFOLDS

(submitted by B. N. Shapukov)

Abstract. The Weil bundle T^{𝔸}M_{n} of an n-dimensional smooth manifold M_{n} determined by a local algebra 𝔸 in the sense of A. Weil carries a natural structure of an n-dimensional 𝔸-smooth manifold. This allows ones to associate with T^{𝔸}M_{n} the series B^{r}(𝔸)T^{𝔸}M_{n}, r = 1, …, ∞, of 𝔸-smooth r-frame bundles. As a set, B^{r}(𝔸)T^{𝔸}M_{n} consists of r-jets of 𝔸-smooth germs of diffeomorphisms (𝔸^{n}, 0) → T^{𝔸}M_{n}. We study the structure of 𝔸-smooth r-frame bundles. In particular, we introduce the structure form of B^{r}(𝔸)T^{𝔸}M_{n} and study its properties.

Next we consider some categories of m-parameter-dependent manifolds whose objects are trivial bundles M_{n} × ℝ^{m} → ℝ^{m}, define (generalized) Weil bundles and higher order frame bundles of m-parameter-de-pendent manifolds and study the structure of these bundles. We also show that product preserving bundle functors on the introduced categories of m-parameter-dependent manifolds are equivalent to generalized Weil functors.

1. Introduction.

The Weil bundle T^{𝔸}M_{n} of a smooth manifold M_{n} corresponding to a local Weil algebra 𝔸 was introduced by A. Weil [40] as a generalization

2000 Mathematical Subject Classification. 58A32, 58A20, 53C05. Key words and phrases. Weil bundle, product preserving bundle functor,

higher order connection.

of the bundle T_{m}^{q}M_{n} of (m, q)-velocities of C. Ehresmann [6]. Various aspects of geometry of Weil bundles were studied by P.C. Yuen [43], [44], L.-N. Patterson [26], A. Morimoto [23], A.P. Shirokov [30], E. Okassa [25], I. Kolář [12], M. Doupovec and I. Kolář [4], A.Ya. Sultanov [35], J. Muñoz, J. Rodriguez, and F.J. Muriel [24] and other researchers.

Studying product preserving functors on the category of smooth manifolds, D.J. Eck [5], G. Kainz and P. Michor [10], and O.O. Luciano [19] proved that such functors reduce to Weil functors T^{𝔸}, which assign to a manifold M_{n} their Weil bundles T^{𝔸}M_{n}. In a series of papers [20], [15], [13] I. Kolář and W.M. Mikulski clarified the structure of product preserving and fiber product preserving bundle functors on the category of fibered manifolds. A. Kriegl and P.W. Michor [16] studied product preserving functors of infinite dimensional manifolds.

A.P. Shirokov [29], [30] discovered that the Weil bundle T^{𝔸}M_{n} carries a natural structure of an n-dimensional manifold over the algebra 𝔸, that is, a manifold whose local coordinates take values in 𝔸 and coordinate transformations are 𝔸-smooth in the sense of G. Scheffers [28]. Natural structures of manifolds over algebras arise also on semitangent bundles studied by V.V. Vishnevsky, [37], [38]. Extensive lists of references on the subject can be found in [30], [14], [37], [38], [34].

The bundle T^{𝔸}M_{n} is naturally associated with the principal q-frame bundle B^{q}M_{n} whose structure group is the differential group G_{n}^{q}, where q is the height (order) of 𝔸. The 𝔸-smooth structure of T^{𝔸}M_{n} implies that T^{𝔸}M_{n} can be considered as a bundle with structure group D_{n}(𝔸), so-called [31], [34] 𝔸-affine differential group. In the case of tangent bundle T M_{n}, this group is the group of affine transformations of ℝ^{n}. In [2], it was shown that D_{n}(𝔸) appears as a natural structure group of the (generalized) Weil bundle T̂^{𝔸}(M_{n}×ℝ^{ℓ}), ℓ is the width of 𝔸, over an object of the category of trivial bundles M_{n} × ℝ^{ℓ} → ℝ^{ℓ} with smooth mappings M_{n} × ℝ^{ℓ} → M′_{k} × ℝ^{ℓ} projecting into the identity mapping id : ℝ^{ℓ} → ℝ^{ℓ} as morphisms. T̂^{𝔸}(M_{n} × ℝ^{ℓ}) → M_{n} × ℝ^{ℓ} is defined as the bundle over M_{n} × ℝ^{ℓ} of 𝔸-velocities of smooth sections ℝ^{ℓ} → M_{n} × ℝ^{ℓ}. Note that Weil bundles of type T^{𝔸}M ×ℝ and natural affinors on such bundles were studied by M. Doupovec and I. Kolář in [4].

This paper is devoted to the study of higher order geometry of Weil bundles considered as manifolds over algebras and to the study of geometry of (generalized) Weil bundles over m-parameter-dependent manifolds M_{n} × ℝ^{m}.

In Section 2, we recall necessary notions and results from the theory of Weil bundles and smooth manifolds over Weil algebras. We also describe here the category Mf^{m} of m-parameter-dependent smooth manifolds and outline some results from [3] concerning the structure of product preserving bundle functors on Mf^{m}.

In Section 3, we study the structure of the bundle B^{r}(𝔸)T^{𝔸}M_{n} of 𝔸smooth r-frames over the Weil bundle T^{𝔸}M_{n}. We construct the structure form Θ^{r} of B^{r}(𝔸)T^{𝔸}M_{n}, study its properties, and derive the structure equations of B^{r}(𝔸)T^{𝔸}M_{n}. In particular, it is proved that a local diffeomorphism of B^{r}(𝔸)T^{𝔸}M_{n} which maps the structure form Θ^{r} into itself coincides in a neighborhood of every point with the prolongation of a local 𝔸-diffeomorphism of T^{𝔸}M_{n}.

In Section 4, we study the structure of product preserving bundle functors on the category Mf_{tr}^{m} whose objects are the trivial fiber bundles M_{n} ×U → U, where M_{n} is a smooth manifold and U is an open subset of ℝ^{m}, and whose morphisms are the smooth mappings M_{n} × U → M′ × U′ which project into translations of ℝ^{m}. It is proved that each such functor is equivalent to a (generalized) Weil functor T̂_{σ}^{𝔸} determined by a Weil algebra 𝔸 and a constant section σ : ℝ^{m} → T^{𝔸}ℝ^{m}.

Section 5 is devoted to the higher order geometry of manifolds from the category Mf_{tr}^{m}. We construct the principal r-frame bundles B̂^{r}(M_{n} ×U) associated with a manifold M_{n} ×U from Mf_{tr}^{m}, define the structure form of B̂^{r}(M_{n} ×ℝ^{m}), prove that a local diffeomorphism of B̂^{r}(M_{n} ×U) which maps the structure form into itself is the natural prolongation of a local isomorphism from the category Mf_{tr}^{m}, and derive the structure equations of B̂^{r}(M_{n} × U). We also construct connections in B̂^{r}(M_{n} × U) and the associated connections in the Weil bundles T̂_{σ}^{𝔸}(M_{n} × U).

2. Preliminaries.

2.1. Weil algebras. A finite-dimensional commutative associative ℝ-al-gebra 𝔸 with unity 1_{𝔸} is called a local Weil algebra or, briefly, a Weil algebra [40], [14], [34] if it has a unique maximal ideal 𝔪 = 𝔪(𝔸) consisting of all nilpotent elements of 𝔸 and the quotient algebra 𝔸/𝔪 is isomorphic to ℝ.

The dimension ℓ of the quotient algebra 𝔪/𝔪^{2} is called the width of 𝔸. The natural number q defined by the relations 𝔪^{q} ≠ 0, 𝔪^{q+1} = 0 is called the height or the order of 𝔸. The ideal 𝔪 is generated by every collection of elements {τ^{a}}, a = 1, …, ℓ, such that the collection of residue classes {τ^{a} + 𝔪^{2}} is a basis of 𝔪/𝔪^{2}. Following V.V. Wagner [39], we will call such a collection {τ^{a}}, a = 1, …, ℓ, a pseudobasis of 𝔪 (and of 𝔸). Every element X of 𝔸 can be represented in the form of a linear combination of products of powers of pseudobasis elements: X = ∑_{|p|=0}^{q} X_{p}τ^{p}, where p = (p_{1}, …, p_{ℓ}) is a multiindex of length ℓ, |p| = p_{1} + ⋯ + p_{ℓ}, τ^{p} = (τ^{1})^{p_{1}} … (τ^{ℓ})^{p_{ℓ}}, τ^{0} = 1_{𝔸}.

A Weil algebra 𝔸 of width ℓ and height q is isomorphic to a quotient algebra of the algebra ℝ[[t^{1}, …, t^{ℓ}]] = ℝ(ℓ, ∞) of formal power series in ℓ variables t^{1}, …, t^{ℓ} with coefficients in ℝ. The mapping φ which assigns to a formal power series ∑_{|p|=0}^{∞} α_{p}t^{p} (as above, p = (p_{1} …, p_{ℓ}) is a multiindex) the element ∑_{|p|=0}^{∞} α_{p}τ^{p} ∈ 𝔸 is an epimorphism of algebras, and 𝔸 ≅ ℝ(ℓ, ∞)/ker φ. The epimorphism of algebras φ : ℝ(ℓ, ∞) → 𝔸 induces the epimorphism of the modules of n-tuples φ : ℝ(ℓ, ∞)^{n} → 𝔸^{n}, which, for simplicity, is denoted by the same symbol φ.

2.2. Weil bundles. Let M_{n} be a smooth (C^{∞} differentiable) manifold, and let f : (ℝ^{ℓ}, 0) → (M_{n}, x) be a smooth germ. In a local chart (U, h) on M_{n}, where h : U ∋ x ↦ {x^{i} = h^{i}(x)} ∈ U′ ⊂ ℝ^{n}, i = 1, …, n, to the jet j^{∞}f of f there corresponds the jet j^{∞}(h ∘ f) which can be considered as the collection of n jets j^{∞}(h^{i} ∘ f) or formal Taylor series of the germs h^{i}∘f. Therefore, j^{∞}(h∘f) can be considered as an element of the module ℝ(ℓ, ∞)^{n}. The jets j^{∞}f and j^{∞}g of germs f, g : (ℝ^{ℓ}, 0) → (M_{n}, x) are said to be 𝔸-equivalent if φ(j^{∞}(h ∘ f)) = φ(j^{∞}(h ∘ g)) or φ(j^{∞}(h ∘ f − h ∘ g)) = 0. 𝔸-equivalence of jets j^{∞}f and j^{∞}g does not depend on the choice of a chart (U, h) and is an equivalence relation on the set J_{0}^{∞}(ℝ^{ℓ}, M_{n}) of ∞-jets of smooth germs from ℝ^{ℓ} to M_{n} at zero (see [34], [14]). The equivalence class of a jet j^{∞}f is called an 𝔸-jet or an 𝔸-velocity on M_{n} and denoted by j^{𝔸}f. On the set T^{𝔸}M_{n} of 𝔸-veloci-ties on M_{n}, there arises a natural structure of a smooth manifold. The natural projection π : T^{𝔸}M_{n} → M_{n} which assigns to the 𝔸-velocity of a germ f : (ℝ^{ℓ}, 0) → (M_{n}, x) the point x ∈ M_{n} turns the manifold T^{𝔸}M_{n} into a fiber bundle over M_{n} called the Weil bundle. A smooth mapping g : M_{n} → M′_{k} induces naturally the mapping of Weil bundles T^{𝔸}g : T^{𝔸}M_{n} → T^{𝔸}M′_{k}, j^{𝔸}f ↦ j^{𝔸}(g ∘ f). The correspondence T^{𝔸} which assigns to a smooth manifold M_{n} the bundle T^{𝔸}M_{n} and to a smooth mapping g the mapping T^{𝔸}g is a functor from the category of smooth manifolds to the category of fibered manifolds. The functor T^{𝔸} is called a Weil functor. The Weil functor T^{𝔸} depends on the choice of a pseudobasis in 𝔸 (or on an epimorphism φ). The choice of another pseudobasis gives an equivalent functor.

A chart (U, h) on a manifold M_{n} induces the mapping h^{𝔸} : π^{−1}(U) → 𝔸^{n}, j^{𝔸}f ↦ φ(j^{∞}(h ∘ f)), which defines an 𝔸^{n}-valued chart (π^{−1}(U), h^{𝔸}) on the bundle T^{𝔸}M_{n}, and the C^{∞} atlas {(U_{α}, h_{α})}_{α∈A} on M_{n} induces the C^{∞} atlas {(π^{−1}(U_{α}), h_{α}^{𝔸})}_{α∈A} on T^{𝔸}M_{n} defining on T^{𝔸}M_{n} a structure of an 𝔸-smooth manifold modeled on the 𝔸-module 𝔸^{n} [34]. In the case of manifolds ℝ and ℝ^{n}, this leads to the natural identifications [34], [33]

T^{𝔸}ℝ ≡ 𝔸, T^{𝔸}ℝ^{n} ≡ 𝔸^{n}.  (1)

In what follows, as a rule, the maximal ideal 𝔪 = 𝔪(𝔸) of a Weil algebra 𝔸 will be denoted by 𝔸^{∘}. By 𝔸^{∘}^{n} we will denote the submodule in 𝔸^{n} consisting of all elements with components belonging to 𝔸^{∘}. On the identifications (1), the fibers T_{0}^{𝔸}ℝ and T_{0}^{𝔸}ℝ^{n} of the bundles T^{𝔸}ℝ and T^{𝔸}ℝ^{n} are identified with the ideal 𝔸^{∘} and the submodule 𝔸^{∘}^{n} respectively.

The mapping h^{𝔸} maps the domain π^{−1}(U) onto U′ × 𝔸^{∘}^{n} ⊂ 𝔸^{n}. As this takes place, the fibers of T^{𝔸}M_{n} are mapped bijectively onto the submodule 𝔸^{∘}^{n}. Thus, T^{𝔸}M_{n} is a locally trivial fiber bundle with standard fiber 𝔸^{∘}^{n}. The bundle T^{𝔸}M_{n} is naturally associated with the q-frame bundle B^{q}M_{n} [42], [14], [34] of M_{n} whose elements are the q-jets of germs of diffeomorphisms (ℝ^{n}, 0) → M_{n}. The structure group of B^{q}M_{n} is the differential group G_{n}^{q} consisting of the q-jets of germs of diffeomorphisms (ℝ^{n}, 0) → (ℝ^{n}, 0). The group G_{n}^{q} acts on the left on 𝔸^{∘}^{n} ≡ T_{0}^{𝔸}ℝ^{n} as follows: if j^{q}φ ∈ G_{n}^{q} and j^{𝔸}f ∈ T_{0}^{𝔸}ℝ^{n} are determined, respectively, by germs φ and f, then

G_{n}^{q} × T_{0}^{𝔸}ℝ^{n} ∋ (j^{q}φ, j^{𝔸}f) ↦ j^{q}φ ∘ j^{𝔸}f = j^{𝔸}(φ ∘ f).

Thus, to the bundle T^{𝔸}M_{n}, there is naturally associated the sequence of principal bundles of higher order frames

M_{n} ⟵^{π_{0}^{1}} B^{1}M_{n} ⟵^{π_{1}^{2}} B^{2}M_{n} ⟵^{π_{2}^{3}}

… ⟵^{π_{q−1}^{q}} B^{q}M_{n} ⟵^{π_{q}^{q+1}} B^{q+1}M_{n} ⟵^{π_{q+1}^{q+2}} … ⟵ B^{∞}M_{n} , (2)

where B^{∞}M_{n} is the bundle of infinite order frames, the limit of the projective system M_{n} ← B^{1}M_{n} ← B^{2}M_{n} ← … endowed with the corresponding structure of an infinite-dimensional smooth manifold in the sense of Bernshtein–Rozenfeld [1]. The bundle B^{∞}M_{n} is formed by the infinite order jets of germs of diffeomorphisms (ℝ^{n}, 0) → M_{n}. The group G_{n}^{∞} consisting of the infinite order jets of germs of diffeomorphisms (ℝ^{n}, 0) → (ℝ^{n}, 0) acts naturally on the right on B^{∞}M_{n}.

2.3. The structure of 𝔸-smooth mappings. Let U ⊂ 𝔸^{n} be an open subset. A smooth mapping F : U ⊂ 𝔸^{n} → 𝔸^{k} is called 𝔸-smooth if the tangent mapping T_{X}F : T_{X}U ≅ 𝔸^{n} → T_{F (X)}𝔸^{k} ≅ 𝔸^{k} is 𝔸-linear for any X ∈ U. Let {e_{a}}, a = 0, 1, …, k, e_{0} = 1_{𝔸} ≡ 1 ∈ ℝ, be a basis in 𝔸, γ_{bc}^{a} the structure constants of 𝔸 with respect to the basis {e_{a}}, and let X^{i} = x^{ia}e_{a}, F^{i′} = f^{i′b}e_{b}, i = 1, …, n, i′ = 1, …, k, be expansions of elements of 𝔸 in terms of {e_{a}} written in accordance with the standard summation convention. A mapping U ∋ {X^{i}} ↦ F^{i′}(X^{i}) = f^{i′b}(x^{ia})e_{b} ∈ 𝔸 is 𝔸-smooth if and only if it satisfies the Scheffers conditions ([28]; [34],(2.6); [33], (10))

∂_{ia}f^{i′b} = γ_{ag}^{b}∂_{i0}f^{i′g}, where ∂_{ia}f^{i′b} = ∂f^{i′b}/∂x^{ia}.  (3)

For an 𝔸-smooth function F (X^{i}), the partial derivatives ∂F/∂X^{i} with respect to the variables X^{i} are defined, and (see [34], [33])

∂F/∂X^{i} = ∂F/∂x^{i0}.  (4)

An arbitrary 𝔸-smooth mapping F : U ⊂ 𝔸^{n} → 𝔸^{k} is of the form [34], [33]

X^{i′} = f^{i′}(x^{i}) + ∑_{|p|=1}^{q} \frac{1}{p!} \frac{D^{p}f^{i′}}{Dx^{p}} X^{∘}^{p},  (5)

where p = (p_{1} …, p_{n}) is a multiindex of length n, X^{∘}^{p} = (X^{∘}^{1})^{p_{1}} … (X^{∘}^{n})^{p_{n}}, X^{∘} ∈ 𝔸^{∘} denotes the component of X ∈ 𝔸 in accordance with the decom-

position 𝔸 = ℝ ⊕ 𝔸^{∘}, and f : U ∋ {x^{i}} ↦ {y^{i′} = f^{i′}(x^{i})} ∈ 𝔸^{k} is a

smooth mapping projectable with respect to the canonical 𝔸^{∘}^{n}-foliation on U generated by the projection π : 𝔸^{n} → ℝ^{n}. If U is a simple open set [22] for the canonical 𝔸^{∘}^{n}-foliation, i. e., the preimages of points from π(U) under the projection π : 𝔸^{n} → ℝ^{n} are connected, then 𝔸-smooth mapping F : U → 𝔸^{k} prolongs uniquely to an 𝔸-smooth mapping

F̃ : π^{−1}(π(U)) → 𝔸^{k}.  (6)

In this case, we have f = (F̃|π(U)) ∘ π.

This implies, in particular, that every 𝔸-smooth germ F : (𝔸^{n}, 0) → 𝔸^{k} is defined along the whole fiber 𝔸^{∘}^{n}, i. e., it is an equivalence class of 𝔸smooth mappings defined on neighborhoods of the form π^{−1}(U), where U are neighborhoods of zero in ℝ^{n}. For any two germs of 𝔸-diffeomorph-

isms F_{1} : (𝔸^{n}, 0) → (𝔸^{n}, X^{∘}), F_{2} : (𝔸^{n}, 0) → (𝔸^{n}, Y^{∘}), where X^{∘}, Y^{∘} ∈ 𝔸^{∘}^{n},

the composition F_{2} ∘ F_{1} : (𝔸^{n}, 0) → 𝔸^{n} is well-defined.

The mapping F : U ⊂ 𝔸^{n} → 𝔸^{k} defined by (5) is called the 𝔸-prolon-gation (analytic prolongation [34]) of f : U → 𝔸^{k}.

Similar relations hold for 𝔸-smooth germs of the form

F : (T^{𝔸}M_{n}, X) → (T^{𝔸}M′_{k}, Y).

Every such a germ is defined along the whole fiber T_{x}^{𝔸}M_{n}, where x = π(X), and is the 𝔸-prolongation of a smooth germ f : (M_{n}, x) → (T^{𝔸}M′_{k}, Y). In terms of local coordinates, the germs f and F are given, respectively, by equations y^{i′} = f^{i′}(x^{i}) and (5).

2.4. The category of m-parameter-dependent manifolds Mf^{m}. In [3], the following category Mf^{m} of m-parameter-dependent manifolds was considered. The objects of Mf^{m} are the trivial fiber bundles p : M_{n}× ℝ^{m} → ℝ^{m}, where M_{n} is a smooth manifold. The morphisms of Mf^{m} are the commutative diagrams of the form

\xymatrix{M_{n} × ℝ^{m} \ar[r]^{f} \ar[d]_{p} & M′_{k} × ℝ^{m} \ar[d]^{p′} \\ ℝ^{m} \ar[r]^{id} & ℝ^{m}}  (7)

In terms of local coordinates (x^{i}, t^{a}), i = 1, …, n, a = 1, …, m, on M_{n} × ℝ^{m} and (x^{i′}, t^{a}), i′ = 1′, …, k′, on M′_{k} × ℝ^{m}, a morphism (7) is given by equations x^{i′} = f^{i′}(x^{i}, t^{a}), t^{a′} = t^{a}.

The category FM^{m} of m-parameter-dependent fibered manifolds is defined as follows. The objects of FM^{m} are the commutative diagrams

\xymatrix{E × ℝ^{m} \ar[r]^{p_{E}} \ar[d]_{π} & ℝ^{m} \ar[d]^{id} \\ M_{n} × ℝ^{m} \ar[r]^{p} & ℝ^{m}}

where p_{E} : E × ℝ^{m} → ℝ^{m} and p : M_{n} × ℝ^{m} → ℝ^{m} are objects of Mf^{m}. The morphisms of FM^{m} are the commutative diagrams

[figure: E × ℝ^{m}; π; f; E′ × ℝ^{m}; π′; M_{n} × ℝ^{m}; p; \overline{f}; M′_{k} × ℝ^{m}; p′; ℝ^{m}; id; id; ℝ^{m}; id; ℝ^{m}; id; ℝ^{m}]  (8)

The base functor

B : FM^{m} → Mf^{m}

is defined as follows:

\xymatrix{E × ℝ^{m} \ar[d] \ar[r] & ℝ^{m} \ar[d] \\ M_{n} × ℝ^{m} \ar[r] & ℝ^{m}} ⟼ (M_{n} × ℝ^{m} → ℝ^{m}), (f, \overline{f}) ↦ \overline{f} .

The functor

ε : FM^{m} → Mf^{m}

which erases the fibered structure is defined as follows:

\xymatrix{E × ℝ^{m} \ar[d] \ar[r] & ℝ^{m} \ar[d] \\ M_{n} × ℝ^{m} \ar[r] & ℝ^{m}} ⟼ (E × ℝ^{m} → ℝ^{m}), (f, \overline{f}) ↦ f .

Note 1. Objects of the category Mf of smooth manifolds can be considered as fiber bundles of the form M_{n} → {pt}, where {pt} = ℝ^{0} is a fixed one-point manifold. With this in mind, we will identify the categories Mf and Mf^{0}.

Every smooth mapping γ : ℝ^{m′} → ℝ^{m}, by means of the pullback construction [14], defines the functor R_{γ} : Mf^{m} → Mf^{m′} which acts on morphisms as follows:

R_{γ}(f(x, t)) = f(x, γ(t)) .

Such functors satisfy the relation

R_{γ_{1}} ∘ R_{γ_{2}} = R_{γ_{2}∘γ_{1}} .  (9)

In particular, the mapping

t : {pt} → ℝ^{m}, {pt} ↦ t ∈ ℝ^{m},

defines the functor R_{t} : Mf^{m} → Mf which acts on objects and morphisms, respectively, as follows: R_{t}(M_{n} × ℝ^{m}) = M_{n}, R_{t}(f : M_{n} × ℝ^{m} → M′_{k} × ℝ^{m}) = (f|_{t} : M_{n} → M′_{k}). The action of these functors on objects is the same for all t ∈ ℝ^{m}:

R_{t}|_{Ob(Mf^{m})} = R_{0}|_{Ob(Mf^{m})} ,

where R_{0} is the functor corresponding to t = 0.

The mapping pt : ℝ^{m} → {pt} defines the functor R_{pt} : Mf → Mf^{m}:

R_{pt}(M_{n}) = M_{n} × ℝ^{m}, R_{pt}(f) = f × id,

which embeds Mf into Mf^{m}.

A covariant functor

F : Mf^{m} → FM^{m}  (10)

satisfying the prolongation condition B ∘ F = id_{Mf^{m}} is called a prolongation functor.

To a prolongation functor (10), one can associate the functor F_{0} : Mf → Mf defined by

F_{0} = R_{0} ∘ ε ∘ F ∘ R_{pt} .

A prolongation functor (10) is called a bundle functor if it satisfies the following localization condition (see [14] for the case of the category Mf): if V ⊂ M_{n} is an open subset and i : V × ℝ^{m} → M_{n} × ℝ^{m} is an Mf^{m}-inclusion, then F_{0}(V) × ℝ^{m} = π_{M}^{−1}(V × ℝ^{m}) and F (i) : π_{M}^{−1}(V × ℝ^{m}) → F_{0}(M_{n}) × ℝ^{m} is an FM^{m}-inclusion (for brevity, we indicate only the upper rows of the diagrams).

The action of a bundle functor F on objects can be written as follows:

F : (M_{n} × ℝ^{m} → ℝ^{m}) ⟼ \xymatrix{F_{0}(M_{n}) × ℝ^{m} \ar[d] \ar[r] & ℝ^{m} \ar[d]^{id} \\ M_{n} × ℝ^{m} \ar[r] & ℝ^{m}}

2.5. Products in the category Mf^{m}. The product of two objects C_{1} and C_{2} of a category 𝒞 is defined [17] to be a triple (P, pr_{1}, pr_{2}) consisting of an object P and two morphisms pr_{1} : P → C_{1}, pr_{2} : P → C_{2} of 𝒞 satisfying the following property: for any object D and any morphisms f_{1} : D → C_{1} and f_{2} : D → C_{2}, there exists a unique morphism f : D → P such that f_{1} = pr_{1} ∘f and f_{2} = pr_{2} ∘f, i. e., the diagram

\xymatrix{C_{1} & P \ar[l]_{^{pr^{1}}} \ar[r]^{^{pr^{2}}} & C_{2} \\  & D \ar[ul]^{f_{1}} \ar@{-->}[u]^{f} \ar[ur]_{f_{2}}}  (11)

is commutative.

In [3], it was shown that the triple (R_{pt}(M_{n} × M′_{k}), R_{pt}(pr_{1}), R_{pt}(pr_{2})) is a product of the objects p_{1} : M_{n} × ℝ^{m} → ℝ^{m} and p_{2} : M′_{k} × ℝ^{m} → ℝ^{m} of the category Mf^{m}.

2.6. Product preserving bundle functors on the category Mf^{m}. Let 𝒞 and 𝒟 be arbitrary categories. A functor F : 𝒞 → 𝒟 is called [14] a product preserving functor if, for any product (A, pr_{1}, pr_{2}) in the category 𝒞, the triple (F (A), F (pr_{1}), F (pr_{2})) is a product in the category 𝒟.

Let Mf × ℝ^{m} denote the subcategory of Mf^{m} whose class of objects coincides with that of the category Mf^{m} and whose morphisms are of the form f × id : M_{n} × ℝ^{m} → M′_{k} × ℝ^{m}, where f : M_{n} → M′_{k} is an arbitrary smooth mapping. The restriction of a bundle functor F : Mf^{m} → FM^{m} to the subcategory Mf × ℝ^{m} is denoted by \overline{F} : Mf × ℝ^{m} → FM^{m}. If F preserves products, then \overline{F} also preserves products.

The following theorem has been proved in [3].

Theorem 1. Let F : Mf^{m} → FM^{m} be a product preserving bundle functor. Then the functor \overline{F} : Mf ×ℝ^{m} → FM^{m} is naturally equivalent to an m-parameter family of Weil functors T̃^{𝔸(t)}.

A product preserving bundle functor F : Mf^{m} → FM^{m} is uniquely determined (up to a natural equivalence) by an m-parameter family of Weil functors T̃^{𝔸(t)} and a collection of functions ℝ^{m} ∋ t ↦ S^{∘} ^{a}(t) ∈ 𝔸^{∘} (t), a = 1, …, m, which gives a section ℝ^{m} → 𝔸(t)^{m} × ℝ^{m}.

For exact definitions of the notions of m-parameter families of Weil algebras and Weil functors, see Section 4.

In addition, in terms of local coordinates, the morphism F (f) has equations of the form

Y^{i} = f^{i}(x, t) + ∑_{p+s=1}^{q} \frac{1}{p!s!} \frac{∂^{|p+s|}f^{i}}{∂x^{p}∂t^{s}} X^{∘}^{p}(S^{∘} (t))^{s}.  (12)

Equations (12) can be rewritten as follows:

Y^{i} = f^{i}(x, t) + ∑_{p=0}^{q} \frac{1}{p!} \frac{∂^{|p|}}{∂x^{p}} \{∑_{s=1}^{q} \frac{1}{s!} \frac{∂^{|s|}f^{i}}{∂t^{s}} (S^{∘} (t))^{s}\} X^{∘}^{p} =

= f̂^{i}(x, t) + ∑_{p=1}^{q} \frac{1}{p!} \frac{∂^{|p|}f̂^{i}}{∂x^{p}} X^{∘}^{p},

where

f̂^{i}(x, t) = f^{i}(x, t) + ∑_{s=1}^{q} \frac{1}{s!} \frac{∂^{|s|}f^{i}}{∂t^{s}} (S^{∘} (t))^{s},

whence it follows that the restriction of F (f) to the fiber over t ∈ ℝ^{m} is an 𝔸(t)-smooth mapping.

3. Principal fiber bundles of 𝔸-smooth frames on T^{𝔸}M_{n}.

3.1. The bundle B^{r}(𝔸)T^{𝔸}M_{n} of 𝔸-smooth r-frames on T^{𝔸}M_{n}. An 𝔸-smooth frame of order r (an 𝔸-smooth r-frame) on T^{𝔸}M_{n} is the rjet of a germ of 𝔸-diffeomorphism Φ : (𝔸^{n}, 0) → (T^{𝔸}M_{n}, X) [32]. The set B^{r}(𝔸)T^{𝔸}M_{n} of 𝔸-smooth r-frames on T^{𝔸}M_{n} is an 𝔸-smooth principal fiber bundle over T^{𝔸}M_{n}, and the canonical projection π^{r}(𝔸) :

B^{r}(𝔸)T^{𝔸}M_{n} → T^{𝔸}M_{n} is an 𝔸-smooth mapping. Since every 𝔸-smooth germ Φ : (𝔸^{n}, 0) → (T^{𝔸}M_{n}, X) is uniquely determined by its restriction φ = Φ|ℝ^{n} : (ℝ^{n}, 0) → (T^{𝔸}M_{n}, X), to an 𝔸-smooth r-frame j^{r}Φ there corresponds bijectively the r-jet j^{r}φ ∈ T^{ℝ(n,r)}T^{𝔸}M_{n}, where ℝ(n, r) is the algebra of truncated polynomials of degree ≤ r in n variables. Taking into account the natural equivalence of the functors T^{ℝ(n,r)} ∘ T^{𝔸} ≅ T^{𝔸} ∘ T^{ℝ(n,r)} ≅ T^{ℝ(n,r)⊗𝔸} [23], [14], [32], we obtain the natural identification B^{r}(𝔸)T^{𝔸}M_{n} ≅ T^{𝔸}(B^{r}M_{n}) and the natural embedding of B^{r}(𝔸)T^{𝔸}M_{n} into T^{ℝ(n,r)⊗𝔸}M_{n} as an open subset, whence it follows that the bundle B^{r}(𝔸)T^{𝔸}M_{n} carries a natural structure of a smooth manifold over the algebra 𝔸 ⊗ ℝ(n, r). Under the identification B^{r}(𝔸)T^{𝔸}M_{n} ≅ T^{𝔸}(B^{r}M_{n}), to the zero section 0 : B^{r}M_{n} → T^{𝔸}(B^{r}M_{n}) there corresponds the natural embedding

ι : B^{r}M_{n} ∋ j_{0}^{r}f ↦ j_{0}^{r}(T^{𝔸}f) ∈ B^{r}(𝔸)T^{𝔸}M_{n},  (13)

where f : (ℝ^{n}, 0) → (M_{n}, x) is a germ of diffeomorphism and T^{𝔸}f : (𝔸^{n}, 0) → (T^{𝔸}M_{n}, x) is the 𝔸-prolongation of f, a germ of 𝔸-diffeo-morphism.

The structure group of the principal bundle π^{r}(𝔸) : B^{r}(𝔸)T^{𝔸}M_{n} → T^{𝔸}M_{n} is the Lie group G_{n}^{r}(𝔸) of r-jets of germs of 𝔸-diffeomorphisms (𝔸^{n}, 0) → (𝔸^{n}, 0), which is isomorphic to the Lie group T^{𝔸}G_{n}^{r} [32].

Thus, in addition to the sequence of principal bundles of higher order frames (2), there is associated to T^{𝔸}M_{n} the sequence of principal bundles of 𝔸-smooth frames

T^{𝔸}M_{n} ⟵^{π_{0}^{1}(𝔸)} B^{1}(𝔸)T^{𝔸}M_{n} ⟵^{π_{1}^{2}(𝔸)} B^{2}(𝔸)T^{𝔸}M_{n} ⟵^{π_{2}^{3}(𝔸)} …

⟵^{π_{r−1}^{r}(𝔸)} B^{r}(𝔸)T^{𝔸}M_{n} ⟵^{π_{r}^{r+1}(𝔸)} … ⟵ B^{∞}(𝔸)T^{𝔸}M_{n} , (14)

where B^{∞}(𝔸)T^{𝔸}M_{n} is the bundle of 𝔸-smooth frames of infinite order, the limit of the projective system T^{𝔸}M_{n} ← B^{1}(𝔸)T^{𝔸}M_{n} ← … endowed with a structure of an infinite-dimensional smooth (and 𝔸smooth) manifold in the sense of Bernshtein–Rozenfeld [1]. Let π_{r}^{∞}(𝔸) : B^{∞}(𝔸)T^{𝔸}M_{n} → B^{r}(𝔸)T^{𝔸}M_{n} denote the canonical projection. The bundle B^{∞}(𝔸)T^{𝔸}M_{n} consists of all infinite order jets of germs of 𝔸diffeomorphisms (𝔸^{n}, 0) → T^{𝔸}M_{n}. The differential group G_{n}^{∞}(𝔸) consisting of all infinite order jets of germs of 𝔸-diffeomorphisms (𝔸^{n}, 0) → (𝔸^{n}, 0) acts naturally on the right on B^{∞}(𝔸)T^{𝔸}M_{n}.

In what follows we will need to consider simultaneously elements of the manifolds T^{𝔸}M_{n} and B^{r}(𝔸)T^{𝔸}M_{n} whose coordinates are elements of the algebras 𝔸 and 𝔸 ⊗ ℝ(n, r) respectively. For this reason, we introduce the following notations for elements of the algebras 𝔸 and 𝔸 ⊗ ℝ(n, r). An element X ∈ 𝔸 has the representation X = x + X^{∘}, where x ∈ ℝ and X^{∘} ∈ 𝔸^{∘} in accordance with the decomposition 𝔸 = ℝ ⊕ 𝔸^{∘}. An element \overline{X} ∈ 𝔸 ⊗ ℝ(n, r) can be represented in the form \overline{X} = x + X^{∘} + X^{∗}, where x ∈ ℝ, X^{∘} ∈ 𝔸^{∘}, and X^{∗} ∈ 𝔸 ⊗ ℝ^{∘}(n, r), in accordance with the decomposition 𝔸 ⊗ ℝ(n, r) = ℝ ⊕ 𝔸^{∘} ⊕ 𝔸 ⊗ ℝ^{∘}(n, r), it can also be represented in the form \overline{X} = x+X^{•}, where x ∈ ℝ and X^{•} = X^{∘}+X^{∗} ∈ 𝔪(𝔸⊗ ℝ(n, r)), in accordance with the decomposition 𝔸 ⊗ ℝ(n, r) = ℝ ⊕ 𝔪(𝔸 ⊗ ℝ(n, r)), 𝔪(𝔸 ⊗ ℝ(n, r)) = 𝔸^{∘} ⊕ 𝔸 ⊗ ℝ^{∘}(n, r). According to the above introduced notations, the coordinates \overline{X}^{i} of an element \overline{X} of T^{𝔸⊗ℝ(n,r)}M_{n} (in particular, of an element of B^{r}(𝔸)T^{𝔸}M_{n}) are represented as follows:

\overline{X}^{i} = x^{i} + X^{∘}^{i} + X^{∗}^{i} = x^{i} + X^{•}^{i}.  (15)

Using the standard basis {ε^{p}}, p = (p_{1}, …, p_{n}), |p| = 0, 1, …, r, in the algebra of truncated polynomials ℝ(n, r), where ε^{p} is the residue class of the monomial t^{p} = (t^{1})^{p_{1}} … (t^{n})^{p_{n}} from the algebra ℝ[[t^{1}, …, t^{n}]] = ℝ(n, ∞), one can also represent an element \overline{X} ∈ 𝔸 ⊗ ℝ(n, r) as follows:

\overline{X} = X + X^{∗} = ∑_{|p|=0}^{r} X_{p} ⊗ ε^{p}, where X = x + X^{∘}, X_{p} ∈ 𝔸. (16)

The expansions of the coordinates \overline{X}^{i} of an element \overline{X} of T^{𝔸⊗ℝ(n,r)}M_{n} corresponding to (16) is of the form

\overline{X}^{i} = X^{i} + X^{∗}^{i} = ∑_{|p|=0}^{r} X_{p}^{i} ⊗ ε^{p}, where X_{p}^{i} ∈ 𝔸.  (17)

3.2. The Lie group G_{n}^{r}(𝔸). By the definitions given above, the group G_{n}^{r}(𝔸) is the fiber (π^{r}(𝔸))^{−1}(0) of the bundle π^{r}(𝔸) : B^{r}(𝔸)T^{𝔸}ℝ^{n} → T^{𝔸}ℝ^{n}. Therefore, the standard coordinates {x^{i}}, i = 1, …, n, on ℝ^{n} induce the globally defined coordinates Z^{∗}^{i} ∈ 𝔸 ⊗ 𝔪(ℝ(n, r)) on G_{n}^{r}(𝔸). In accordance with (17), the coordinates Z^{∗}^{i} can be represented in the form

Z^{∗}^{i} = ∑_{|p|=1}^{r} Z_{p}^{i} ⊗ ε^{p}, where Z_{p}^{i} ∈ 𝔸, det(Z_{j}^{i}) ∉ 𝔸^{∘}.  (18)

In terms of these coordinates, the composition Y = X ∘ Z in G_{n}^{r}(𝔸) is written in the form

Y^{∗}^{i} = ∑_{|p|=1}^{r} X_{p}^{i} Z^{∗}^{p}.  (19)

The same formula (19) also gives the right action

R_{Z} : B^{r}(𝔸)T^{𝔸}M_{n} ∋ \overline{X} ↦ \overline{Y} = R_{Z}(\overline{X}) ∈ B^{r}(𝔸)T^{𝔸}M_{n}  (20)

of G_{n}^{r}(𝔸) on B^{r}(𝔸)T^{𝔸}M_{n} in terms of the local coordinates \overline{X}^{i} = x^{i} + X^{∘}^{i} + X^{∗}^{i} on B^{r}(𝔸)T^{𝔸}M_{n} induced by local coordinates x^{i} on M_{n}.

Proposition 1. i) The Lie group G_{n}^{r}(𝔸) is isomorphic to the Lie group of 𝔸-linear automorphisms of the algebra 𝔸 ⊗ ℝ(n, r).

ii) The Lie algebra 𝔤_{n}^{r}(𝔸) of G_{n}^{r}(𝔸) is isomorphic to the Lie algebra of 𝔸-linear derivations of the algebra 𝔸 ⊗ ℝ(n, r) with bracket

[D_{1}, D_{2}] = D_{2} ∘ D_{1} − D_{1} ∘ D_{2}.  (21)

Proof. Passing to the r-jets in the composition F ∘ G of 𝔸-smooth germs G : (𝔸^{n}, 0) → (𝔸^{n}, 0) and F : (𝔸^{n}, 0) → T^{𝔸}M′_{k}, where M′_{k} is an arbitrary k-dimensional smooth manifold, we obtain, similarly to (20), the right action of G_{n}^{r}(𝔸) on T^{𝔸⊗ℝ(n,r)}M′_{k}:

T^{𝔸⊗ℝ(n,r)}M′_{k} × G_{n}^{r}(𝔸) ∋ \{\overline{X}, Z\} ↦ \overline{Y} = R_{Z}(\overline{X}) ∈ T^{𝔸⊗ℝ(n,r)}M′_{k}. (22)

In terms of the local 𝔸 ⊗ ℝ(n, r)-coordinates \overline{X}^{i′} = x^{i′} + X^{∘}^{i′} + X^{∗}^{i′} on T^{𝔸⊗ℝ(n,r)}M′_{k} induced by local coordinates x^{i′} on M′_{k}, this action is given by the equations (19): Y^{∗}^{i′} = ∑_{|p|=1}^{r} X_{p}^{i′} Z^{∗}^{p}. In particular, taking as M′_{k} the field of real numbers, we obtain the right action of G_{n}^{r}(𝔸) on T^{𝔸⊗ℝ(n,r)}ℝ ≅ 𝔸 ⊗ ℝ(n, r):

𝔸 ⊗ ℝ(n, r) × G_{n}^{r}(𝔸) ∋ \{\overline{X}, Z\} ↦ \overline{Y} = R_{Z}(\overline{X}) ∈ 𝔸 ⊗ ℝ(n, r), (23)
{x + X^{∘} + X^{∗}, Z^{∗}^{i}} ↦ {x + X^{∘} + ∑_{|p|=1}^{r} X_{p} Z^{∗}^{p}}.  (23′)

Let \overline{X} = j^{r}F_{1}, \overline{Y} = j^{r}F_{2}, and Z = j^{r}G, where F_{1} : (𝔸^{n}, 0) → 𝔸, F_{2} : (𝔸^{n}, 0) → 𝔸 and G : (𝔸^{n}, 0) → (𝔸^{n}, 0) are 𝔸-smooth germs. Then (F_{1}+F_{2})∘G = F_{1}∘G+F_{2}∘G and (F_{1}·F_{2})∘G = (F_{1}∘G)·(F_{2}∘G). Passing to r-jets, we obtain (\overline{X} + \overline{Y}) ∘ Z = \overline{X} ∘ Z + \overline{Y} ∘ Z and (\overline{X} · \overline{Y}) ∘ Z = (\overline{X} ∘ Z) · (\overline{Y} ∘ Z). Thus, the action (23) of Z ∈ G_{n}^{r}(𝔸) on 𝔸 ⊗ ℝ(n, r) is an automorphism of 𝔸 ⊗ ℝ(n, r). A fixed element X ∈ 𝔸 ⊂ ℝ(n, q) ⊗ 𝔸 can be considered as the r-jet of the constant mapping F_{1} : (𝔸^{n}, 0) → 𝔸. In this case, the relation (F_{1} · F_{2}) ∘ G = (F_{1} ∘ G) · (F_{2} ∘ G) takes the form (X · F_{2}) ∘ G = X · (F_{2} ∘ G). Therefore, (X · \overline{Y}) ∘ Z = X · (\overline{Y} ∘ Z), and so R_{Z} is an 𝔸-linear automorphism. This fact also follows from (23′).

Let now h : 𝔸 ⊗ ℝ(n, r) → 𝔸 ⊗ ℝ(n, r) be an arbitrary 𝔸-linear automorphism of 𝔸 ⊗ ℝ(n, r). The ideal 𝔸^{∘} of 𝔸, considered as a subset of 𝔸 ⊗ ℝ(n, r), generates the ideal 𝔸^{∘} ⊗ ℝ(n, r) of 𝔸 ⊗ ℝ(n, r). The quotient algebra 𝔸 ⊗ ℝ(n, r)/ 𝔸^{∘} ⊗ ℝ(n, r) can be identified with ℝ(n, r). The ideal 𝔸^{∘} ⊗ ℝ(n, r) is invariant under 𝔸-linear automorphisms of 𝔸 ⊗ ℝ(n, r). Therefore, the automorphism h induces an automorphism of ℝ(n, r). Hence it follows that h acts on the generators ε^{i}, i = 1, …, n of the algebra ℝ(n, r) ⊂ 𝔸 ⊗ ℝ(n, r) being the residue classes of the monomials t^{i}, i = 1, …, n, as follows:

h : ε^{i} ↦ ∑_{|p|=1}^{r} α_{p}^{i} ⊗ ε^{p} + α^{i}, where α_{p}^{i} ∈ 𝔸, det(α_{j}^{i}) ∉ 𝔸^{∘}, α^{i} ∈ 𝔸^{∘}. (24)

Denote ε̃^{i} = ∑_{|p|=1}^{r} α_{p}^{i} ⊗ ε^{p} and α̃^{i} = ε̃^{i} + α^{i}. Since ε̃^{p} = 0 and α̃^{p} = 0

for |p| > r, it follows that, in (24), α^{i} = 0. Thus, h(ε^{i}) = Z^{∗}^{i} ∈ 𝔸 ⊗ 𝔪(ℝ(n, r)) and, for an arbitrary \overline{X} = ∑_{|p|=0}^{r} X_{p} ⊗ ε^{p} ∈ ℝ(n, q) ⊗ 𝔸,

X_{p} ∈ 𝔸, we have h(\overline{X}) = ∑_{|p|=0}^{r} X_{p}h(ε^{p}) = ∑_{|p|=0}^{r} X_{p} Z^{∗}^{p} = R_{Z}(\overline{X}),

where Z = {Z^{∗}^{i}} ∈ G_{n}^{r}(𝔸) ⊂ (𝔸 ⊗ 𝔪(ℝ(n, q))^{n}. Thus, h = R_{Z}, and the first statement has been proved.

To prove the second statement, we need to find the fundamental vector fields of the action of G_{n}^{r}(𝔸) on T^{𝔸⊗ℝ(n,r)}M′_{k}.

Consider first a more general situation. Let F : T^{𝔸}M_{n} → T^{𝔸}M′_{k} be an 𝔸-smooth mapping. The composition of the r-jet j^{r}F with r-jets of 𝔸smooth germs (𝔸^{n}, 0) → T^{𝔸}M_{n} defines the 𝔸 ⊗ ℝ(n, r)-smooth mapping

F^{𝔸⊗ℝ(n,r)} : T^{𝔸⊗ℝ(n,r)}M_{n} → T^{𝔸⊗ℝ(n,r)}M′_{k}.  (25)

In terms of the local coordinates induced by local coordinates on M_{n} and M′_{k}, the mapping F^{𝔸⊗ℝ(n,r)} has the form (5):

\overline{X}^{i′} = f^{i′}(x^{j}) + ∑_{|p|=1}^{r+q} \frac{1}{p!} \frac{D^{p}f^{i′}(x^{j})}{Dx^{p}} X^{•}^{p},  (26)

where f^{i′}(x^{j}) are the 𝔸-valued functions which give the restriction of F to the zero section M_{n} ⊂ T^{𝔸}M_{n}. Using the partial derivatives of the 𝔸-smooth functions

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

which give the mapping F in terms of local 𝔸-coordinates, one can rewrite (26) in the form

\overline{X}^{i′} = \overline{F}^{i′}(\overline{X}^{j}) = F^{i′}(X^{j}) + ∑_{|p|=1}^{r} \frac{1}{p!} \frac{D^{p}F^{i′}(X^{j})}{DX^{p}} X^{∗}^{p},  (27)

where

\frac{D^{p}F^{i′}(X^{j})}{DX^{p}} = \frac{D^{p}F^{i′}(X^{j})}{Dx^{p}} = \frac{D^{p}f^{i′}(x^{j})}{Dx^{p}} + ∑_{|s|=1}^{q} \frac{1}{s!} \frac{D^{p+s}f^{i′}(x^{j})}{Dx^{p+s}} X^{∘}^{p}.

Let \overline{X} ∈ T^{𝔸⊗ℝ(n,r)}M_{n}, and let \overline{X}′ ∈ T^{𝔸⊗ℝ(n,r)}M′_{k} be the image of \overline{X} under the mapping (25). The mapping

T F^{𝔸⊗ℝ(n,r)} : T T^{𝔸⊗ℝ(n,r)}M_{n} → T T^{𝔸⊗ℝ(n,r)}M′_{k}  (28)

of the tangent bundles induced by (25), in terms of local coordinates, is given by the equations \overline{X}^{i′} = \overline{F}^{i′}(\overline{X}^{j}) of the form (27) and by the equations

\overline{V}^{i′} = \frac{∂\overline{F}^{i′}}{∂\overline{X}^{j}} \overline{V}^{j} = (\frac{∂F^{i′}}{∂X^{j}} + ∑_{|p|=1}^{r} \frac{1}{p!} \frac{D^{p+j}F^{i′}}{DX^{p+j}} X^{∗}^{p}) \overline{V}^{j}.  (29)

The coordinates \overline{V}^{j} of a tangent vector \overline{V} ∈ T\overline{X}T^{𝔸⊗ℝ(n,r)}M_{n}, as well as the coordinates of elements of T^{𝔸⊗ℝ(n,r)}M_{n}, are of the form \overline{V}^{j} = V^{j} +V^{∗}^{j},

where V^{j} ∈ 𝔸 and V^{∗}^{j} ∈ 𝔸 ⊗ 𝔪(ℝ(n, r)).

If V^{j} = 0, then the elements V^{∗}^{j} are the coordinates of a vertical vector on the bundle T^{𝔸⊗ℝ(n,r)}M_{n} → T^{𝔸}M_{n}, i. e., of a vector tangent to the fiber T_{X}^{𝔸⊗ℝ(n,r)}M_{n}. In the case V^{j} = 0, the summands in (29) corresponding to |p| = r + 1 are equal to zero. Therefore, the mapping V T\overline{X}T^{𝔸⊗ℝ(n,r)}M_{n} of the vertical tangent spaces is defined by the r-jet j_{X}^{r}F.

Letting now in (28) \overline{X} be the identity e = j_{0}^{r}(id_{𝔸^{n}}) of the Lie group G_{n}^{r}(𝔸) ⊂ T_{0}^{ℝ(n,r)}𝔸^{n} ⊂ T^{𝔸⊗ℝ(n,r)}ℝ^{n}, and \overline{Y} be an arbitrary element

\overline{Y} = j_{0}^{r}F, F : (𝔸^{n} ≅ T^{𝔸}ℝ^{n}, 0) → (T^{𝔸}M′_{k}, Y),

of the bundle T^{𝔸⊗ℝ(n,r)}M′_{k}, we obtain the mapping

T_{e}F^{𝔸⊗ℝ(n,r)} : T_{e}G_{n}^{r}(𝔸) ∋ \overline{V}_{e} ↦ \overline{V} \overline{Y} ∈ T\overline{Y} T^{𝔸⊗ℝ(n,r)}M′_{k}  (30)

which assigns to an element \overline{V}_{e} of the Lie algebra 𝔤_{n}^{r}(𝔸) ≅ T_{e}G_{n}^{r}(𝔸) the vector \overline{V} \overline{Y} being the value at \overline{Y} of the fundamental vector field \overline{V} of the action of G_{n}^{r}(𝔸) on T^{𝔸⊗ℝ(n,r)}M′_{k} corresponding to \overline{V}_{e}. To find the equations of a fundamental vector field \overline{V} on T^{𝔸⊗ℝ(n,r)}M′_{k} in terms of local coordinates, we let X^{∗}^{p} = ε^{p} in equations (29), which determine the mapping (30). Then we obtain

\overline{V}^{i′}(\overline{Y}) = (Y_{j}^{i′} + ∑_{|p|=1}^{r−1}(p_{j} + 1)Y_{p+j}^{i′}ε^{p})\overline{V} _{e}^{j},  (31)

where Y_{j}^{i′}, Y_{p+j}^{i′} ∈ 𝔸, and it is supposed that the sum is taken over the index j. In particular, the fundamental vector fields on B^{r}(𝔸)T^{𝔸}M_{n} are of the form

\overline{V}^{i}(\overline{X}) = (X_{j}^{i} + ∑_{|p|=1}^{r−1}(p_{j} + 1)X_{p+j}^{i}ε^{p})\overline{V} _{e}^{j} = (∑_{|p|=0}^{r−1}(p_{j} + 1)X_{p+j}^{i}ε^{p})\overline{V} _{e}^{j}. (32)

In the case M′_{k} = ℝ, k = 1, formula (30) determines the fundamental vector fields of the action of the group G_{n}^{r}(𝔸) on the algebra 𝔸 ⊗ ℝ(n, r). Let \overline{V}_{e} ∈ T_{e}G_{n}^{r}(𝔸) be an element of the Lie algebra 𝔤_{n}^{r}(𝔸), and let γ : (ℝ, 0) ∋ t ↦ γ(t) ∈ G_{n}^{r}(𝔸) be a germ of curve such that (dγ/dt)|_{0} = \overline{V}_{e}. By (30), the value of the fundamental vector field \overline{V} on 𝔸 ⊗ ℝ(n, r) corresponding to \overline{V}_{e} at α = j^{r}F ∈ 𝔸 ⊗ ℝ(n, r), F : (𝔸^{n}, 0) → 𝔸, is of the form

\overline{V} (α) = \overline{V}_{α} = T_{e}j^{r}F (\overline{V}_{e}) = \frac{d}{dt}|_{0} (j^{r}F ∘ γ) = \frac{d}{dt}|_{0} (F^{𝔸⊗ℝ(n,r)} ∘ γ). (33)

Applying (33) to the product α_{1} · α_{2} ∈ 𝔸 ⊗ ℝ(n, r), we obtain

\overline{V} (α_{1} · α_{2}) = \frac{d}{dt}|_{0} ((F_{1}^{𝔸⊗ℝ(n,r)} · F_{2}^{𝔸⊗ℝ(n,r)}) ∘ γ) =

= \frac{d}{dt}|_{0} (F_{1}^{𝔸⊗ℝ(n,r)} ∘ γ) · (F_{2}^{𝔸⊗ℝ(n,r)} ∘ γ)(0)+

+ (F_{1}^{𝔸⊗ℝ(n,r)} ∘ γ)(0) · \frac{d}{dt}|_{0} (F_{2}^{𝔸⊗ℝ(n,r)} ∘ γ) = \overline{V} (α_{1}) · α_{2} + α_{1} · \overline{V} (α_{2}).

If α_{1} ∈ 𝔸 ⊂ 𝔸 ⊗ ℝ(n, r), then α_{1} is the r-jet α_{1} = j^{r}F_{1} of the constant germ F_{1} : (𝔸^{n}, 0) ↦ α_{1} ∈ 𝔸. Consequently, \overline{V} (α_{1}) = 0 and \overline{V} (α_{1} · α_{2}) = α_{1} · \overline{V} (α_{2}). Now, by means of similar calculations, one can easily show that, for β_{1}, β_{2} ∈ 𝔸, α_{1}, α_{2} ∈ 𝔸 ⊗ ℝ(n, r), the following relation holds: \overline{V} (β_{1}α_{1} + β_{2}α_{2}) = β_{1}\overline{V} (α_{1}) + β_{2}\overline{V} (α_{2}).

An arbitrary 𝔸-linear derivation D : 𝔸 ⊗ ℝ(n, r) → 𝔸 ⊗ ℝ(n, r) is determined by its values D(ε^{i}) ∈ 𝔪(𝔸 ⊗ ℝ(n, r)) on the generators ε^{i}, i = 1, …, n, of the algebra ℝ(n, r) ⊂ 𝔸 ⊗ ℝ(n, r). Since D(ε^{p}) = 0 for |p| = r + 1, it follows (as in the case of automorphisms (24) of the algebra 𝔸⊗ℝ(n, r)) that D(ε^{i}) = \overline{V}^{i} = ∑_{|p|=1}^{r} V_{p}^{i}⊗ε^{p}, where V_{p}^{i} ∈ 𝔸, i. e., \overline{V}^{i} ∈ 𝔸⊗𝔪(ℝ(n, r)). Therefore, D coincides with the derivation which is the fundamental vector field \overline{V} generated by the element \overline{V}_{e} ∈ T_{e}G_{n}^{r}(𝔸) = T_{e}(𝔸 ⊗ 𝔪(ℝ(n, r))) ≅ 𝔸 ⊗ 𝔪(ℝ(n, r)) with 𝔸 ⊗ ℝ(n, r)-coordinates \overline{V}^{i}. In fact, the 𝔸 ⊗ ℝ(n, r)-coordinates of the identity e ∈ G_{n}^{r}(𝔸) are the elements \{ε^{i}\}, i = 1, …, n, and the values of the coordinates \overline{V}^{i}(e) of the left-invariant vector field corresponding to \overline{V}_{e} ∈ T_{e}G_{n}^{r}(𝔸) at e ∈ G_{n}^{r}(𝔸), by (33), are equal to \overline{V}^{i}(e) = \overline{V} (ε^{i}).

To the Lie bracket of fundamental vector fields on 𝔸 ⊗ ℝ(n, r) there corresponds the Lie bracket of derivations (21). In fact, the group G_{n}^{r}(𝔸), which acts on the right on 𝔸 ⊗ 𝔪(ℝ(n, r)) by linear transformations, and the space T_{e}G_{n}^{r}(𝔸) can be considered as subsets in the automorphism algebra End(𝔸 ⊗ 𝔪(ℝ(n, r))) of the vector space 𝔸 ⊗ 𝔪(ℝ(n, r)) acting on the right on 𝔸 ⊗ 𝔪(ℝ(n, r)). □

3.3. The Lie bracket of 𝔸-smooth vector fields. The tangent bundle T 𝔸^{n} of the 𝔸-module 𝔸^{n} is naturally identified with 𝔸^{n}×𝔸^{n}, and if {X^{i}}, i = 1, …, n, are the standard 𝔸-coordinates on 𝔸^{n}, then a vector field on an open subset O ⊂ 𝔸^{n} is given by a smooth mapping V : O ∋ {X^{i}} ↦ {X^{i}, V^{j}(X^{i})} ∈ 𝔸^{n} × 𝔸^{n}. A vector field V is 𝔸-smooth if the functions V^{j} = V^{j}(X^{i}) are 𝔸-smooth. In order that a vector field V be 𝔸-smooth it is necessary and sufficient that the functions v^{jb} = v^{jb}(x^{ia}) satisfy the conditions (3):

∂_{ia}v^{jb} = γ_{ag}^{b}∂_{i0}v^{jg}, where ∂_{ia}v^{jb} = ∂v^{jb}/∂x^{ia}.  (34)

Proposition 2. Let U and V be 𝔸-smooth vector fields on a domain O ⊂ 𝔸^{n}, and let W = [U, V] be the Lie bracket of these fields. Then

i) W is an 𝔸-smooth vector field.

ii) The 𝔸-coordinates W^{j} of W are of the form

W^{j} = U^{i}∂_{i}V^{j} − V^{i}∂_{i}U^{j}.  (35)

Proof. Statement i) follows from statement ii) since the functions U^{j}, V^{j} and their partial derivatives are 𝔸-smooth.

Let us prove statement ii). By the definition of the Lie bracket of vector fields, the conditions of 𝔸-smoothness of a vector field (34), and by relation (4) for partial derivatives of an 𝔸-smooth function, we have

W^{j} = w^{jb}e_{b} =

= (u^{ia}∂_{ia}v^{jb} − v^{ia}∂_{ia}u^{jb})e_{b} = (u^{ia}γ_{ag}^{b}∂_{i0}v^{jg} − v^{ia}γ_{ag}^{b}∂_{i0}u^{jg})e_{b} =

= u^{ia}e_{a}∂_{i0}v^{jg}e_{g} − v^{ia}e_{a}∂_{i0}u^{jg}e_{g} = U^{i}∂_{i}V^{j} − V^{i}∂_{i}U^{j}. (36)

□

3.4. Lie algebras of germs and jets of 𝔸-smooth vector fields. We will use the following notation:

Vect_{0}(ℝ^{n}) is the Lie algebra of germs of vector fields on ℝ^{n} at zero, Vect_{0}(ℝ^{n}, 0) is the Lie algebra of germs of vector fields on ℝ^{n} at zero

which take zero value at zero.

Proposition 2 allows one to consider also the following Lie algebras of germs of 𝔸-smooth vector fields:

Vect_{0}(𝔸^{n}), the Lie algebra of germs of 𝔸-smooth vector fields on 𝔸^{n} at zero,

Vect_{0}(𝔸^{n}, 0), the Lie algebra of germs of 𝔸-smooth vector fields on 𝔸^{n} at zero which take zero value at zero,

Vect_{0}(𝔸^{n}, 𝔸^{∘}^{n}), the Lie algebra of germs of 𝔸-smooth vector fields on 𝔸^{n} at zero whose value at zero belongs to the submodule 𝔸^{∘}^{n}.

Vect_{0}(𝔸^{n}, 𝕀^{n}), the Lie algebra of germs of 𝔸-smooth vector fields on 𝔸^{n} at zero whose value at zero belongs to the submodule 𝕀^{n} ∈ 𝔸^{n} consisting of the n-tuples with elements from an ideal 𝕀 of 𝔸.

Proposition 3. The following Lie algebra isomorphisms take place:

Vect_{0}(𝔸^{n}) ≅ 𝔸 ⊗ Vect_{0}(ℝ^{n}),  (37)
Vect_{0}(𝔸^{n}, 0) ≅ 𝔸 ⊗ Vect_{0}(ℝ^{n}, 0).  (38)

Proof. The isomorphisms (37) and (38) follow from the fact that the germs of 𝔸-smooth vector fields U, V, and [U, V] on 𝔸^{n} at zero are completely defined by their restrictions to ℝ^{n} (see (36)). The restriction u = U|ℝ^{n} of a germ of 𝔸-smooth vector field U is a germ of 𝔸^{n}-valued vector field on ℝ^{n} at zero. The germ u is of the form u^{j}(x^{i}) = u^{jb}(x^{i})e_{b}. Since 𝔸 is a finite-dimensional algebra, u can be considered as an element e_{b} ⊗ u^{jb}(x^{i}) of the Lie algebra 𝔸 ⊗ Vect_{0}(ℝ^{n}). The germ of 𝔸-smooth vector field U is restored from u as its 𝔸-prolongation (see (5)) U^{j} = u^{j}(x^{i}) + ∑_{|p|=1}^{q} \frac{1}{p!}(D^{p}u^{i}/Dx^{p}) X^{∘}^{p}. Thus, the isomorphism of Lie algebras 𝔸 ⊗ Vect_{0}(ℝ^{n}) → Vect_{0}(𝔸^{n}) is realized by passing from germs of 𝔸^{n}-va-lued vector fields to their 𝔸-prolongations. □

Note that the isomorphism (37) allows ones to regard the Lie algebras Vect_{0}(𝔸^{n}, 𝔸^{∘}^{n}) and Vect_{0}(𝔸^{n}, 𝕀^{n}) as Lie subalgebras in 𝔸 ⊗ Vect_{0}(ℝ^{n}).

Passing in the Lie algebras Vect_{0}(ℝ^{n}) and Vect_{0}(ℝ^{n}, 0) from germs of vector fields at zero to the ∞-jets, one obtains the following Lie algebras [1], [7]:

Vect_{0}^{∞}(ℝ^{n}) = W_{n}, the Lie algebra of formal vector fields on ℝ^{n}, an element V ∈ W_{n} is a collection of n formal power series V^{i} = ∑_{|p|=0}^{∞} v_{p}^{i}t^{p} ∈ ℝ[[t^{1}, …, t^{n}]], i = 1, …, n, the Lie bracket [U, V] of two formal vector fields is computed by formula (35), where ∂_{i} = ∂/∂t^{i} is the operator of formal differentiation of a power series with respect to t^{i}, for this reason, V ∈ W_{n} can also be written as the linear combination V^{i}∂_{i};

Vect_{0}^{∞}(ℝ^{n}, 0) = L_{0}(W_{n}), the Lie subalgebra in W_{n} of formal vector fields V with V^{i} belonging to the maximal ideal 𝔪(ℝ[[t^{1}, …, t^{n}]]) of the algebra ℝ[[t^{1}, …, t^{n}]], which consists of the series with zero constant term, i. e., such that V^{i} = ∑_{|p|=1}^{∞} v_{p}^{i}t^{p};

L_{r}(W_{n}), r ≥ 0, the Lie subalgebra in W_{n} of formal vector fields V with V^{i} belonging to the (r + 1)-st power of the ideal 𝔪(ℝ[[t^{1}, …, t^{n}]]), i. e., such that V^{i} = ∑_{|p|=r+1}^{∞} v_{p}^{i}t^{p}.

[L_{r}(W_{n}), L_{s}(W_{n})] ⊂ L_{r+s}(W_{n}), therefore, for r > s, L_{r}(W_{n}) is an ideal in L_{s}(W_{n}). In particular, L_{r}(W_{n}), r > 0, is an ideal in L_{0}(W_{n}).

Passing in the Lie algebras Vect_{0}(𝔸^{n}), Vect_{0}(𝔸^{n}, 0), Vect_{0}(𝔸^{n}, 𝔸^{∘}^{n}), and Vect_{0}(𝔸^{n}, 𝕀^{n}) from germs of 𝔸-smooth vector fields at zero to their ∞- jets, we obtain, respectively, the Lie algebras Vect_{0}^{∞}(𝔸^{n}), Vect_{0}^{∞}(𝔸^{n}, 0), Vect_{0}^{∞}(𝔸^{n}, 𝔸^{∘}^{n}), and Vect_{0}^{∞}(𝔸^{n}, 𝕀^{n}).

Denote by 𝔸[[t^{1}, …, t^{n}]] = 𝔸(n, ∞) the algebra of formal power series α = ∑_{|p|=0}^{∞} α_{p}t^{p}, α_{p} ∈ 𝔸, with coefficients in a Weil algebra 𝔸. Since 𝔸 is finite-dimensional, the algebra 𝔸[[t^{1}, …, t^{n}]] is isomorphic to the tensor product 𝔸 ⊗ ℝ[[t^{1}, …, t^{n}]]. The algebra 𝔸[[t^{1}, …, t^{n}]] has the unique maximal ideal 𝔪(𝔸[[t^{1}, …, t^{n}]]) consisting of the series whose constant term α_{0} belongs to the maximal ideal 𝔸^{∘} of 𝔸.

By the Lie algebra of formal vector fields on 𝔸^{n} we will mean the Lie algebra W_{n}^{𝔸} of formal vector fields V = V^{i}∂_{i} with V^{i} ∈ 𝔸[[t^{1}, …, t^{n}]], i = 1, …, n, with Lie bracket [U, V] defined by (35), where, as in the case of the Lie algebra W_{n}, ∂_{i} = ∂/∂t^{i} is the operator of formal differentiation of formal power series with respect to t^{i}.

There are the following Lie subalgebras in W_{n}^{𝔸}: L_{r}(W_{n}^{𝔸}), r = 0, 1, …, the Lie subalgebra of formal vector fields V = V^{i}∂_{i} such that all the truncated series ∑_{|p|=0}^{r} v_{p}^{i}t^{p} of V^{i} = ∑_{|p|=0}^{∞} v_{p}^{i}t^{p}, i = 1, …, n, are zero; in particular, L_{0}(W_{n}^{𝔸}) is the Lie subalgebra of formal vector fields V with V^{i} having zero constant terms; L̃_{0}(W_{n}^{𝔸}), the Lie subalgebra of formal vector fields V with V^{i} belonging to the maximal ideal 𝔪(𝔸[[t^{1}, …, t^{n}]]). From (35) it follows that L_{r}(W_{n}^{𝔸}), r = 1, …, is an ideal in L_{0}(W_{n}^{𝔸}).

The following propositions are consequences of Propositions 2 and 3.

Proposition 4. The following Lie algebra isomorphisms take place:

Vect_{0}^{∞}(𝔸^{n}) ≅ W_{n}^{𝔸} ≅ 𝔸 ⊗ W_{n},  (39)
Vect_{0}^{∞}(𝔸^{n}, 0) ≅ L_{0}(W_{n}^{𝔸}) ≅ 𝔸 ⊗ L_{0}(W_{n}),  (40)
Vect_{0}^{∞}(𝔸^{n}, 𝔸^{∘}^{n}) ≅ L̃_{0}(W_{n}^{𝔸}).  (41)

Proposition 5. The vector space Vect_{0}^{r}(𝔸^{n}, 0) of r-jets of germs of 𝔸smooth vector fields from Vect_{0}(𝔸^{n}, 0) with bracket [j^{r}V_{1}, j^{r}V_{2}] = j^{r}[V_{1}, V_{2}] is a Lie algebra isomorphic to the quotient Lie algebra

L_{0}(W_{n}^{𝔸})/L_{r+1}(W_{n}^{𝔸}) ≅ 𝔸 ⊗ (L_{0}(W_{n})/L_{r+1}(W_{n})) .

3.5. Lifts of vector fields to B^{r}(𝔸)T^{𝔸}M_{n}. A vector field v : M_{n} → T M_{n} on a smooth manifold M_{n} generates a local flow φ_{t} : M_{n} → M_{n}. The Weil functor T^{𝔸} applied to the flow φ_{t} gives the flow T^{𝔸}φ_{t} : T^{𝔸}M_{n} → T^{𝔸}M_{n} on the Weil bundle T^{𝔸}M_{n}, the 𝔸-prolongation of the flow φ_{t}. The flow T^{𝔸}φ_{t}, in turn, generates the vector field v^{C} : T^{𝔸}M_{n} → T T^{𝔸}M_{n} on T^{𝔸}M_{n} called the complete lift of v. The vector field v^{C} can be obtained as the composition of the 𝔸-prolongation v^{𝔸} : T^{𝔸}M_{n} → T^{𝔸}T M_{n} of the section v : M_{n} → T M_{n} with the diffeomorphism T^{𝔸}T M_{n} → T T^{𝔸}M_{n} which follows from the natural equivalence of functors T^{𝔸}∘T^{ℝ(ε)} ≅ T^{ℝ(ε)}∘ T^{𝔸}. In what follows we will not distinguish between the mappings v^{C} and v^{𝔸}. The local 𝔸-coordinates V^{i}(X^{j}) of the vector field v^{𝔸} on T^{𝔸}M_{n} in terms of the 𝔸-chart h^{𝔸} generated by a chart h on M_{n} are the 𝔸-pro-longations of the coordinates v^{i}(x^{j}) of v. In accordance with (5),

V^{i} = v^{i}(x^{j}) + ∑_{|p|=1}^{q} \frac{1}{p!} \frac{D^{p}v^{i}}{Dx^{p}} X^{∘}^{p}.  (42)

The complete lift \overline{V} of an 𝔸-smooth vector field V on the Weil bundle T^{𝔸}M_{n} to the bundle T^{ℝ(n,r)}T^{𝔸}M_{n} ≅ T^{𝔸⊗ℝ(n,r)}M_{n} is given by a formula similar to (42). If v = V |M_{n} : M_{n} → T T^{𝔸}M_{n} is the restriction of an 𝔸smooth vector field V to M_{n} ⊂ T^{𝔸}M_{n} (we identify M_{n} with zero section of T^{𝔸}M_{n}), then the complete lift \overline{V} = V^{𝔸⊗ℝ(n,r)} of V to T^{𝔸⊗ℝ(n,r)}M_{n}, in terms of local coordinates \{\overline{X}^{i}\} on T^{𝔸⊗ℝ(n,r)}M_{n} generated by local coordinates {x^{i}} on M_{n} is of the form

\overline{V} ^{i}(\overline{X}^{j}) = v^{i}(x^{j}) + ∑_{|p|=1}^{q} \frac{1}{p!} \frac{D^{p}v^{i}}{Dx^{p}} X^{•}^{p} = V^{i}(X^{j}) + ∑_{|p|=1}^{q} \frac{1}{p!} \frac{D^{p}V^{i}}{Dx^{p}} X^{∗}^{p}, (43)

where v^{i}(x^{j}) are the 𝔸-valued smooth functions being the restrictions to ℝ^{n} of the 𝔸-smooth functions V^{i}(X^{j}), X^{•}^{i} ∈ 𝔪(𝔸⊗ℝ(n, r)), and \overline{V} ^{i}(\overline{X}^{j}) are the 𝔸 ⊗ ℝ(n, r)-smooth functions being the prolongations of v^{i}(x^{j}) and V^{i}(X^{j}). The principal bundle B^{r}(𝔸)T^{𝔸}M_{n} is an open submanifold in T^{𝔸⊗ℝ(n,r)}M_{n}. Therefore, formulas (43) give also the complete lift of an 𝔸-smooth vector field V from T^{𝔸}M_{n} to B^{r}(𝔸)T^{𝔸}M_{n}.

Proposition 6. The complete lift \overline{V} of an 𝔸-smooth vector field V from T^{𝔸}M_{n} to B^{r}(𝔸)T^{𝔸}M_{n} is invariant with respect to the right action of the Lie group G_{n}^{r}(𝔸) on B^{r}(𝔸)T^{𝔸}M_{n}.

Proof. In terms of local coordinates, the right action (20)

R_{Z} : B^{r}(𝔸)T^{𝔸}M_{n} ∋ \overline{X} ↦ \overline{Y} = R_{Z}(\overline{X}) ∈ B^{r}(𝔸)T^{𝔸}M_{n}  (44)

is of the form (19):

Y^{∗}^{i} = ∑_{|p|=1}^{r} X_{p}^{i} Z^{∗}^{p},  (45)

where \overline{X}^{i} = x^{i} + X^{∘}^{i} + X^{∗}^{i}, X^{∗}^{i} = X_{p}^{i} ⊗ ε^{p}, \overline{Y}^{i} = x^{i} + X^{∘}^{i} + Y^{∗}^{i} (see (15)).

Then the tangent mapping

T\overline{X}R_{Z} : T\overline{X}B^{r}(𝔸)T^{𝔸}M_{n} ∋ \overline{V} ↦ \overline{W} ∈ T\overline{Y} B^{r}(𝔸)T^{𝔸}M_{n}  (46)

is of the form

W^{∗}^{i} = ∑_{|p|=1}^{r} V_{p}^{i} Z^{∗}^{p},  (47)

where V^{∗}^{i} = V_{p}^{i} ⊗ ε^{p}. The complete lift \overline{V} of an 𝔸-smooth vector field V from T^{𝔸}M_{n} to B^{r}(𝔸)T^{𝔸}M_{n}, in terms of local coordinates, is given by equations (43)

V^{∗}^{i}(\overline{X}^{j}) = ∑_{|p|=1}^{r} V_{p}^{i} X^{∗}^{p}.  (48)

Under the right action (46), the complete lift \overline{V} goes to the vector field \overline{W} whose coordinates W^{∗}^{i} at \overline{Y} = \overline{X} ∘ Z are obtained by substituting Z^{∗}^{i} in place of ε^{i} in the expansion of V^{∗}^{i}(\overline{X}^{j}) from (48) in powers of ε^{p}: V^{∗}^{i}(\overline{X}^{j}) = A_{p}^{i}ε^{p}. For this, one should replace ε^{p} by Z^{∗}^{p} in the expansion of each X^{∗}^{i} in the right-hand side of (48), which is equivalent to the

replacement in (48) of ∑_{|p|=1}^{r} V_{p}^{i} X^{∗}^{p} by ∑_{|p|=1}^{r} V_{p}^{i} Y^{∗}^{p}, where \overline{Y} = \overline{X} ∘ Z,

which, in turn, gives V^{∗}^{i}(\overline{Y} ^{j}). □

Proposition 7. The Lie algebra of right-invariant vector fields on the Lie group G_{n}^{r}(𝔸) is isomorphic to the Lie algebra Vect_{0}^{r}(𝔸^{n}, 0).

Proof. Let V ∈ Vect_{0}(𝔸^{n}, 0) be a germ of 𝔸-smooth vector field at 0 ∈ 𝔸^{n} such that V (0) = 0. In terms of the coordinates X^{i} on 𝔸^{n}, the germ V is given by functions V^{i}(X^{j}). The 𝔸 ⊗ ℝ(n, r)-lift \overline{V} of the germ V at Z ∈ G_{n}^{r}(𝔸) ⊂ B^{r}(𝔸)𝔸^{n}, in terms of the induced coordinates, is of the form (48):

\overline{V}^{i}(Z) = ∑_{|p|=1}^{r} V_{p}^{i} Z^{∗}^{p}, V_{p}^{i} = \frac{1}{p!} \frac{D^{p}V^{i}}{DX^{p}} .  (49)

By Proposition 6, the restriction of the germ \overline{V} to the Lie group G_{n}^{r}(𝔸) is a right-invariant vector field Ṽ on this group. From (49) it follows that the vector field Ṽ is uniquely determined by the r-jet of the germ V. The correspondence (49) assigning to the r-jet of a germ of 𝔸-smooth vector field V the right-invariant vector field Ṽ is bijective since the value of the field Ṽ at the identity e ∈ G_{n}^{r}(𝔸) is of the form Ṽ^{i}(e) = ∑_{|p|=1}^{r} V_{p}^{i}ε^{p}. What is more, the correspondence (49) is a Lie algebra isomorphism. In fact, the Lie bracket [Ũ, Ṽ] of right-invariant vector fields Ũ and Ṽ is the restriction to G_{n}^{r}(𝔸) of the Lie bracket [\overline{U}, \overline{V}] of the complete lifts of germs of 𝔸-smooth vector fields U and V, which coincides with the complete lift \overline{[U, V]} of the Lie bracket [U, V], and the bracket [j_{0}^{r}U, j_{0}^{r}V] in the Lie algebra Vect_{0}^{r}(𝔸^{n}, 0), by the definition, equals to the r-jet j_{0}^{r}[U, V]. □

3.6. Fundamental semivector fields on the bundle B^{r}(𝔸)T^{𝔸}M_{n}. Denote by T^{r_{1}}B^{r}(𝔸)T^{𝔸}M_{n}, 0 ≤ r_{1} ≤ r, the inverse image of the tangent bundle T B^{r_{1}}(𝔸)T^{𝔸}M_{n} under the projection π_{r_{1}}^{r}(𝔸) : B^{r}(𝔸)T^{𝔸}M_{n} → B^{r_{1}}(𝔸)T^{𝔸}M_{n}. An element \overline{V} \overline{X} ∈ T r_{1} ; \overline{X}B^{r}(𝔸)T^{𝔸}M_{n} can be considered as a tangent vector to B^{r}(𝔸)T^{𝔸}M_{n} defined up to a summand belonging to the kernel of the projection T\overline{X}π_{r_{1}}^{r}(𝔸) or, in terms of the algebra ℝ(n, r), up to a summand belonging to the submodule

ℝ^{∘}(n, r)^{r_{1}+1} T\overline{X}B^{r}(𝔸)T^{𝔸}M_{n}

generated by the (r_{1}+1)-st power ℝ^{∘}(n, r)^{r_{1}+1} of the maximal ideal ℝ^{∘}(n, r) of the algebra ℝ(n, r). Therefore, the fiber T r_{1} ; \overline{X}B^{r}(𝔸)T^{𝔸}M_{n} of the bundle T^{r_{1}}B^{r}(𝔸)T^{𝔸}M_{n} → B^{r}(𝔸)T^{𝔸}M_{n} can be considered as the quotient module T\overline{X}B^{r}(𝔸)T^{𝔸}M_{n}/ ℝ^{∘}(n, r)^{r_{1}+1}T\overline{X}B^{r}(𝔸)T^{𝔸}M_{n}. From this point of view, the bundle T^{r_{1}}B^{r}(𝔸)T^{𝔸}M_{n} is the quotient bundle of the vector bundle T B^{r}(𝔸)T^{𝔸}M_{n} by the subbundle ℝ^{∘}(n, r)^{r_{1}+1}T B^{r}(𝔸)T^{𝔸}M_{n} generated by the ideal ℝ^{∘}(n, r)^{r_{1}+1}.

In a similar manner, one can define the quotient bundles

T^{r_{1}}T^{ℝ(n,r)}T^{𝔸}M_{n} → T^{ℝ(n,r)}T^{𝔸}M_{n}.

Sections of the bundles

T^{r_{1}}B^{r}(𝔸)T^{𝔸}M_{n} → B^{r}(𝔸)T^{𝔸}M_{n}

and

T^{r_{1}}T^{ℝ(n,r)}T^{𝔸}M_{n} → T^{ℝ(n,r)}T^{𝔸}M_{n}

will be called, following the terminology of A.M. Vasiliev [36], semivector fields on B^{r}(𝔸)T^{𝔸}M_{n} and T^{ℝ(n,r)}T^{𝔸}M_{n} respectively.

An 𝔸-smooth mapping F : T^{𝔸}M_{n} → T^{𝔸}M′_{k} induces the 𝔸 ⊗ ℝ(n, r)- smooth mapping (25)

F^{𝔸⊗ℝ(n,r)} : T^{𝔸⊗ℝ(n,r)}M_{n} → T^{𝔸⊗ℝ(n,r)}M′_{k},
j_{0,X}^{r}φ ↦ j_{X}^{r}F ∘ j_{0,X}^{r}φ = j_{0,F (X)}^{r}(F ∘ φ),

where φ : (ℝ^{n}, 0) → (T^{𝔸}M_{n}, X), and, in particular, as the restriction of (25), the mapping

F^{𝔸⊗ℝ(n,r)} : B^{r}(𝔸)T^{𝔸}M_{n} → T^{𝔸⊗ℝ(n,r)}M′_{k},

The mapping (25), in turn, induces the mapping of the tangent bundles (28)

T F^{𝔸⊗ℝ(n,r)} : T T^{𝔸⊗ℝ(n,r)}M_{n} → T T^{𝔸⊗ℝ(n,r)}M′_{k}.

If there is given only the jet j_{X}^{r}F of F at X ∈ T^{𝔸}M_{n}, then the mapping

T\overline{X}F^{𝔸⊗ℝ(n,r)} : T\overline{X}T^{𝔸⊗ℝ(n,r)}M_{n} → T\overline{Y} T^{𝔸⊗ℝ(n,r)}M′_{k},
where \overline{Y} = F^{𝔸⊗ℝ(n,r)}(\overline{X}), π(𝔸)_{0}^{r}(\overline{X}) = X.

is not defined, but we have uniquely defined (by equations (29)) the mapping

j_{X}^{r}F = T r−1 ; \overline{X}F^{𝔸⊗ℝ(n,r)} : T r−1 ; \overline{X}T^{𝔸⊗ℝ(n,r)}M_{n} → T r−1 ; \overline{Y}T^{𝔸⊗ℝ(n,r)}M′_{k},

\overline{Y} = F^{𝔸⊗ℝ(n,r)}(\overline{X}), π(𝔸)_{0}^{r}(\overline{X}) = X.

In particular, the jet Y = j_{0}^{r}F of 𝔸-smooth germ F : (𝔸^{n}, 0) → (T^{𝔸}M′_{k}, X) defines the mapping

j_{0}^{r}F : T_{e}^{r−1}B^{r}(𝔸)T^{𝔸}ℝ^{n} → T r−1 ; \overline{Y}T^{𝔸⊗ℝ(n,r)}M′_{k},  (50)

where e is the identity of G_{n}^{r}(𝔸). The subspaces T_{e}^{r−1}B^{r}(𝔸)T^{𝔸}ℝ^{n} and T r−1 ; \overline{Y}T^{𝔸⊗ℝ(n,r)}M′_{k} carry structures of 𝔸 ⊗ ℝ(n, r)-modules isomorphic, respectively, to the modules 𝔸 ⊗ ℝ(n, r − 1)^{n} and 𝔸 ⊗ ℝ(n, r − 1)^{k}, and the mapping (50) is 𝔸 ⊗ ℝ(n, r)-linear. Fixing \overline{V} _{e}^{(r−1)} ∈ T_{e}^{r−1}B^{r}(𝔸)T^{𝔸}ℝ^{n} and varying \overline{Y}, we obtain a semivector field \overline{V}^{(r−1)} on T^{𝔸⊗ℝ(n,r)}M′_{k}.

This semivector field will be called the fundamental semivector field (cf [31]) on T^{𝔸⊗ℝ(n,r)}M′_{k} corresponding to \overline{V} _{e}^{(r−1)}. Similarly, fixing \overline{V} _{e}^{(r′)} ∈ T_{e}^{r′}B^{r}(𝔸)T^{𝔸}ℝ^{n} for r′ < r and varying \overline{Y}, we obtain the fundamental semivector field on T^{𝔸⊗ℝ(n,r)}M′_{k} corresponding to \overline{V} _{e}^{(r′)}.

In terms of local coordinates, fundamental semivector fields are given by equations similar to (31). The semivector field corresponding to \overline{V} _{e}^{(r−1)} = \overline{V} _{e}^{j}∂_{j}(e), \overline{V} _{e}^{j} ∈ 𝔸 ⊗ ℝ(n, r − 1), is given by the equations

\overline{V}^{i′}(\overline{Y}) = (Y_{j}^{i′} + ∑_{|p|=1}^{r−1}(p_{j} + 1)Y_{p+j}^{i′}ε^{p})\overline{V} _{e}^{j},  (51)

where the product in relations of the form (Y_{p+j}^{i′}ε^{p})\overline{V} _{e}^{j} is understood as the action of the algebra 𝔸 ⊗ ℝ(n, r) on the quotient algebra 𝔸 ⊗ ℝ(n, r − 1).

In particular, fundamental semivector fields on the bundle B^{r}(𝔸)T^{𝔸}M_{n} are of the form (see (32))

\overline{V}^{i}(\overline{X}) = (X_{j}^{i} + ∑_{|p|=1}^{r−1}(p_{j} + 1)X_{p+j}^{i}ε^{p})\overline{V} _{e}^{j} = (∑_{|p|=0}^{r−1}(p_{j} + 1)X_{p+j}^{i}ε^{p})\overline{V} _{e}^{j}. (52)

Note 2. The bundles T^{𝔸⊗ℝ(n,r)}M′_{k} and T^{r′}B^{r}(𝔸)T^{𝔸}M^{n} are, obviously, 𝔸smooth manifolds modeled on 𝔸-modules of type 𝔸^{m} for certain m ∈ ℕ (these bundles can also be considered as 𝔸 ⊗ ℝ(n, r)-smooth manifolds modeled on the corresponding 𝔸 ⊗ ℝ(n, r)-modules, see [33]). From (51) and (52) it follows that fundamental semivector fields are 𝔸-smooth sections of these bundles.

Note 3. The complete lift of a germ of 𝔸-smooth vector field V on 𝔸^{n} ≡ T^{𝔸}ℝ^{n} at zero to the bundle B^{r}(𝔸)T^{𝔸}ℝ^{n} is a germ of 𝔸 ⊗ ℝ(n, r)-smooth vector field \overline{V} on B^{r}(𝔸)T^{𝔸}ℝ^{n} defined along the whole fiber G_{n}^{r}(𝔸). This germ has equations (43). If there is given only the (r −1)-jet j_{0}^{r−1}V of V, then formulas (43) define a unique element \overline{V}_{e} ∈ T_{e}^{r−1}B^{r}(𝔸)T^{𝔸}ℝ^{n}, which, in turn, determines a unique (r − 1)-jet j_{0}^{r−1}V. Thus, there is an isomorphism between the 𝔸 ⊗ ℝ(n, r)-modules Vect_{0}^{r−1}(𝔸^{n}) and T_{e}^{r−1}B^{r}(𝔸)𝔸^{n}. Since the bundle B^{r}(𝔸)𝔸^{n} is an open submanifold of T^{𝔸⊗ℝ(n,r)}ℝ^{n} ≡ (𝔸⊗ ℝ(n, r))^{n}, the 𝔸 ⊗ ℝ(n, r)-module T_{e}^{r−1}B^{r}(𝔸)𝔸^{n} is canonically isomorphic to the 𝔸 ⊗ ℝ(n, r)-module (𝔸 ⊗ ℝ(n, r − 1))^{n}. From (51) it follows that the set of all fundamental semivector fields on T^{𝔸⊗ℝ(n,r)}M′_{k} is also an 𝔸 ⊗ ℝ(n, r − 1)-module isomorphic to (𝔸⊗ℝ(n, r −1))^{n}. We will denote the 𝔸 ⊗ ℝ(n, r′)-module of fundamental semivector fields on T^{𝔸⊗ℝ(n,r)}M′_{k} by T^{r′}T^{𝔸⊗ℝ(n,r)}M′_{k}.

In the case M′_{k} = ℝ, k = 1, formula (51) determines the fundamental semivector fields on the algebra 𝔸 ⊗ ℝ(n, r). As in the proof of Proposition 1, one can easily verify (see also [31]) that these fundamental semivector fields are 𝔸-linear derivations from the algebra 𝔸 ⊗ ℝ(n, r) to the algebra 𝔸 ⊗ ℝ(n, r − 1) with respect to the canonical epimorphism π_{r−1}^{r} : 𝔸 ⊗ ℝ(n, r) → 𝔸 ⊗ ℝ(n, r − 1), and that the set 𝔇_{r−1}^{r}(𝔸) of all such derivations coincides with the set of all fundamental semivector fields. By Note 3, 𝔇_{r−1}^{r}(𝔸) is an 𝔸 ⊗ ℝ(n, r − 1)-module isomorphic to (𝔸 ⊗ ℝ(n, r − 1))^{n}. The 𝔸-linear derivations ∂_{j} : 𝔸 ⊗ ℝ(n, r) → 𝔸 ⊗ ℝ(n, r − 1) defined by ∂_{j}(ε^{i}) = δ_{j}^{i} form a basis in this module. Using the derivations ∂_{j}, one can rewrite (51) as follows

\overline{V}^{i′}(\overline{Y}) = \overline{V} _{e}^{j}∂_{j}(\overline{Y}^{i′}), where \overline{Y}^{i′} = ∑_{|p|=0}^{r} Y_{p}^{i′} ⊗ ε^{p}, Y_{p}^{i′} ∈ 𝔸. (53)

Relations (53) establish an isomorphism between the 𝔸 ⊗ ℝ(n, r − 1)- module T_{e}^{r−1}B^{r}(𝔸)T^{𝔸}ℝ^{n} and the 𝔸 ⊗ ℝ(n, r − 1)-module of fundamental semivector fields on T^{𝔸⊗ℝ(n,r)}M′_{k}. In particular, fundamental semivector fields on 𝔸 ⊗ ℝ(n, r) are of the form

\overline{V} (α) = \overline{V}^{j}∂_{j}(α), \overline{V}^{j} ∈ 𝔸 ⊗ ℝ(n, r − 1).  (54)

Note 4. The multiplication in 𝔸 ⊗ ℝ(n, r) induces the action

𝔸 ⊗ ℝ(n, r − 1) × 𝔸 ⊗ 𝔪(ℝ(n, r)) → 𝔸 ⊗ 𝔪(ℝ(n, r)).

With this in mind, we can express an arbitrary fundamental vector field on 𝔸 ⊗ ℝ(n, r) (𝔸-linear derivation of 𝔸 ⊗ ℝ(n, r)) in the same form as (54):

\overline{V} (α) = \overline{V}^{j}∂_{j}(α), \overline{V}^{j} ∈ 𝔸 ⊗ 𝔪(ℝ(n, r)),  (55)

where the operators ∂_{j} have the same sense as in (54).

Denote by 𝔇_{r′}^{r}(𝔸), r′ < r, the 𝔸 ⊗ ℝ(n, r − 1)-module of 𝔸-linear derivations from 𝔸 ⊗ ℝ(n, r) to 𝔸 ⊗ ℝ(n, r′) with respect to the canonical epimorphism π_{r′}^{r} : 𝔸 ⊗ ℝ(n, r) → 𝔸 ⊗ ℝ(n, r′). Each derivation D ∈ 𝔇_{r−1}^{r}(𝔸) generates a series of derivations D_{r′}^{r} ∈ 𝔇_{r′}^{r}(𝔸), r′ < r − 1, defined by D_{r′}^{r} = π_{r′}^{r−1} ∘ D. We will also denote D_{r′}^{r} simply by D. Then the relation [D_{1}, D_{2}] = D_{2} ∘ D_{1} − D_{1} ∘ D_{2} defines the bracket

[ ,] : 𝔇_{r−1}^{r}(𝔸) × 𝔇_{r−1}^{r}(𝔸) → 𝔇_{r−2}^{r}(𝔸).  (56)

In terms of the derivations ∂_{j}, the bracket (56) is completely determined by the relations

[∂_{i}, ∂_{j}] = 0,  (57)

from which it follows that if [\overline{V} _{e}^{(r−1)}, \overline{W} _{e}^{(r−1)}] = \overline{U}_{e}^{(r−2)}, then

\overline{U}_{e}^{i}∂_{i} = [\overline{V} _{e}^{j}∂_{j}, \overline{W} _{e}^{j}∂_{j}] = (\overline{W} _{e}^{j}∂_{j}(\overline{V} _{e}^{i}) − \overline{V} _{e}^{j}∂_{j}(\overline{W} _{e}^{i}))∂_{i}.  (58)

It will be convenient to identify \overline{V} _{e}^{(r−1)} = \{\overline{V} _{e}^{j}\} ∈ T_{e}^{r−1}B^{r}(𝔸)ℝ^{n} with \overline{V} _{e}^{j}∂_{j} ∈ 𝔇_{r−1}^{r}(𝔸).

The Lie bracket of vector fields on T^{𝔸⊗ℝ(n,r)}M′_{k} induces the 𝔸-linear bracket of fundamental semivector fields

[ ,] : T^{r−1}T^{𝔸⊗ℝ(n,r)}M′_{k} × T^{r−1}T^{𝔸⊗ℝ(n,r)}M′_{k} → T^{r−2}T^{𝔸⊗ℝ(n,r)}M′_{k}, (59)

which corresponds to the bracket (56) since the fundamental semivector fields (53) on T^{𝔸⊗ℝ(n,r)}ℝ^{k} can be regarded as 𝔸-linear mappings (cf the proof of Proposition 1).

3.7. The structure form of the bundle B^{r}(𝔸)T^{𝔸}M_{n}. The structure form θ^{r} of the bundle B^{r}M_{n} of r-frames on a real smooth manifold M_{n} is a 1-form on B^{r}M_{n} with values in the space T_{e}^{r−1}B^{r}ℝ^{n}, where e is the identity of the differential group G_{n}^{r} ⊂ B^{r}ℝ^{n}. The form θ^{r} is defined as follows [42]. If X = j^{r}f, where f : (ℝ^{n}, 0) → (M_{n}, x) is a germ of diffeomorphism, then

θ_{X}^{r}(v_{X}) = j^{r}(f^{−1})(T π_{r−1}^{r}(v_{X})).

As the domain of values of the structure form θ^{r}, one can also take the space Vect_{0}^{r−1}(ℝ^{n}) of (r − 1)-jets of vector fields at zero on ℝ^{n} [9], [21].

In a similar way, we define the T_{e}^{r−1}B^{r}(𝔸)T^{𝔸}ℝ^{n}-valued (or, equivalently, (𝔸 ⊗ ℝ(n, r − 1))^{n}-valued) structure form Θ^{r} = Θ^{r}(T^{𝔸}M_{n}) of the bundle B^{r}(𝔸)T^{𝔸}M_{n}. Let \overline{X} = j^{r}F, where F : (𝔸^{n}, 0) → (T^{𝔸}M_{n}, X) is a germ of 𝔸-diffeomorphism, and let \overline{V} \overline{X} be a tangent vector to B^{r}(𝔸)T^{𝔸}M_{n} at \overline{X}. We let

Θr ; \overline{X}(\overline{V} \overline{X}) = j^{r}(F^{−1})(T π_{r−1}^{r}(\overline{V} \overline{X})).  (60)

From (60) it follows that in fact the form Θ^{r} is defined on elements from T^{r−1}B^{r}(𝔸)T^{𝔸}M_{n} and assigns to the value \overline{V} (r−1) ; \overline{X} at \overline{X} of the fundamental semivector field \overline{V}^{(r−1)} corresponding to an element \overline{V} _{e}^{r−1} the same element \overline{V} _{e}^{r−1}.

Proposition 8. The structure form Θ^{r} is an 𝔸-smooth mapping.

Proof. In terms of local coordinates, a fundamental semivector field on B^{r}(𝔸)T^{𝔸}M_{n} is given by functions \overline{V}^{i}(\overline{X}) = (∑_{|p|=0}^{r} \overline{X}_{p+j}^{i})\overline{V} _{e}^{j} (see (52)) of 𝔸-valued coordinates \overline{X}_{p}^{i}, which, obviously, are 𝔸-smooth. When the elements \overline{V} _{e}^{j} ∈ 𝔸⊗ℝ(n, r −1) run through the all possible values, equations (52) define an 𝔸-smooth trivialization

ψ : (𝔸 ⊗ ℝ(n, r − 1))^{n} × B^{r}(𝔸)T^{𝔸}M_{n} → T^{r−1}B^{r}(𝔸)T^{𝔸}M_{n}  (61)

of the bundle T^{r−1}B^{r}(𝔸)T^{𝔸}M_{n}. Using the mappings

λ : (𝔸 ⊗ ℝ(n, r − 1))^{n} ∋ \{\overline{V}^{j}\}_{e}∂_{j}(e) ↦ \overline{V} _{e}^{j}∂_{j}(e) ∈ T_{e}^{r−1}B^{r}(𝔸)𝔸^{n}

and

pr_{1} : (𝔸 ⊗ ℝ(n, r − 1))^{n} × B^{r}(𝔸)T^{𝔸}M_{n} ↦ (𝔸 ⊗ ℝ(n, r − 1))^{n},

we can represent the structure form Θ^{r} as the composition

Θ^{r} = λ ∘ pr_{1} ∘ ψ^{−1} ∘ π_{r−1}^{r}.

Since ψ is an 𝔸-diffeomorphism, its inverse ψ^{−1} is also an 𝔸-diffeomor-phism, whence it follows that the form Θ^{r} is 𝔸-smooth. □

The structure form Θ^{r} possesses the following property, which is an analogue of the corresponding property of the structure form of the bundle of r-frames of real smooth manifold M_{n} (see, e. g., [21]).

Theorem 2. Let \overline{Φ} : B^{r}(𝔸)T^{𝔸}M_{n} → B^{r}(𝔸)T^{𝔸}M′_{n} be a local diffeomorphism which maps the structure form Θ^{r} of the bundle B^{r}(𝔸)T^{𝔸}M_{n} into the the structure form Θ′^{r} of the bundle B^{r}(𝔸)T^{𝔸}M′_{n}. Then in a neighborhood of every point \overline{X} ∈ B^{r}(𝔸)T^{𝔸}M_{n} the diffeomorphism \overline{Φ} coincides with the 𝔸 ⊗ ℝ(n, r)-prolongation of a local 𝔸-diffeomorphism Φ : T^{𝔸}M_{n} → T^{𝔸}M′_{n}.

Proof. In the proof of this statement, we will use the scheme applied in [21] (Section 1.3.1). We will suppose that the structure forms take values in the 𝔸 ⊗ ℝ(n, r)-module (𝔸 ⊗ ℝ(n, r − 1))^{n}.

1) If \overline{Φ} coincides with 𝔸 ⊗ ℝ(n, r)-prolongation Φ^{𝔸⊗ℝ(n,r)} of a local 𝔸-diffeomorphism Φ : T^{𝔸}M_{n} → T^{𝔸}M′_{n}, then \overline{Φ} maps the fundamental semivector field on B^{r}(𝔸)T^{𝔸}M_{n} corresponding to \overline{V} _{e}^{(r−1)} ∈ T^{r−1}B^{r}(𝔸)𝔸^{n} into the fundamental semivector field on B^{r}(𝔸)T^{𝔸}M′_{n} corresponding to the same element \overline{V} _{e}^{(r−1)}. Hence it follows that \overline{Φ} maps the structure form of B^{r}(𝔸)T^{𝔸}M_{n} into the structure form of B^{r}(𝔸)T^{𝔸}M′_{n}.

2) Since the structure form Θ^{r} establishes an isomorphism between the 𝔸 ⊗ ℝ(n, r)-modules T r−1 ; \overline{X}B^{r}(𝔸)T^{𝔸}M_{n} and (𝔸 ⊗ ℝ(n, r − 1))^{n}, it also establishes an isomorphism between the submodules of these modules generated by the ideals 𝔸 ⊗ (ℝ^{∘}(n, r − 1))^{r_{1}}, r_{1} = 0, 1, …, r − 1. Hence it follows that \overline{Φ} is fibered over the local diffeomorphisms

\overline{Φ}^{r_{1}} : B^{r_{1}}(𝔸)T^{𝔸}M_{n} → B^{r_{1}}(𝔸)T^{𝔸}M′_{n}, r_{1} = 0, 1, …, r − 1.

In particular, \overline{Φ} is fibered over Φ = \overline{Φ}^{0} : T^{𝔸}M_{n} → T^{𝔸}M′_{n}.

3) In the case r = 1, the local diffeomorphism \overline{Φ}^{1} : B^{1}(𝔸)T^{𝔸}M_{n} → B^{1}(𝔸)T^{𝔸}M′_{n} is fibered over Φ : T^{𝔸}M_{n} → T^{𝔸}M′_{n}. Since the map \overline{Φ}^{1} induces an isomorphism of 𝔸-modules T 0 ; \overline{X}B^{1}(𝔸)T^{𝔸}M_{n} ≡ T_{X}T^{𝔸}M_{n} and T_{\overline{Φ}(\overline{X})}^{0}B^{1}(𝔸)T^{𝔸}M′_{n} ≡ T_{Φ(X)}T^{𝔸}M′_{n}, the tangent mappings T_{X}Φ are 𝔸-lin-ear. Hence it follows that the mapping Φ : T^{𝔸}M_{n} → T^{𝔸}M′_{n} is 𝔸-smooth.

4) Assume now that the statement of the theorem holds for the bundles of 𝔸-smooth p-frames for p = 1, …, r − 1. As has been shown in item 2), the mapping \overline{Φ} is fibered over \overline{Φ}^{r−1} : B^{r−1}(𝔸)T^{𝔸}M_{n} → B^{r−1}(𝔸)T^{𝔸}M′_{n}. Since fundamental semivector fields which are sections of the bundle T^{r−2}B^{r}(𝔸)T^{𝔸}M_{n} → B^{r}(𝔸)T^{𝔸}M_{n} project into fundamental semivector fields being sections of the bundle T^{r−2}B^{r−1}(𝔸)T^{𝔸}M_{n} → B^{r−1}(𝔸)T^{𝔸}M_{n}, it follows that the mapping \overline{Φ}^{r−1} maps the structure form of B^{r−1}(𝔸)T^{𝔸}M_{n} into the structure form of B^{r−1}(𝔸)T^{𝔸}M′_{n}. Hence it follows that the mapping \overline{Φ}^{r−1} coincides with the 𝔸 ⊗ ℝ(n, r − 1)- prolongation of a local 𝔸-diffeomorphism Φ : T^{𝔸}M_{n} → T^{𝔸}M′_{n}. Consider the 𝔸 ⊗ ℝ(n, r)-prolongation Φ^{𝔸⊗ℝ(n,r)} of Φ and the composition Ψ = (Φ^{𝔸⊗ℝ(n,r)})^{−1} ∘ \overline{Φ} : B^{r}(𝔸)T^{𝔸}M_{n} → B^{r}(𝔸)T^{𝔸}M_{n}. The local diffeomorphism Ψ preserves the structure form and projects into the identity diffeomorphism of the bundle B^{r−1}(𝔸)T^{𝔸}M_{n}. Therefore, Ψ is a family of right translations of the bundle B^{r}(𝔸)T^{𝔸}M_{n} over the bundle B^{r−1}(𝔸)T^{𝔸}M_{n}. Such right translations Z ∈ G_{n}^{r}(𝔸) have coordinates of the form Z_{k}^{i} = δ_{k}^{i}, Z_{p}^{i} = 0 for |p| ≤ r − 1, Z_{p}^{i} ∈ 𝔸 for |p| = r and, therefore, do not change the coordinates of elements of the bundle T^{r−1}B^{r}(𝔸)T^{𝔸}M_{n}. Equating the coordinates (52) of the values of the fundamental semivector field corresponding to an arbitrary element \overline{V} _{e}^{(r−1)} ∈ T_{e}^{r−1}B^{r}(𝔸)𝔸^{n} at the points \overline{X} and R\overline{Z}(\overline{X}), namely,

\overline{V}^{i}(\overline{X}) = (∑_{|p|=0}^{r−1}(p_{j} + 1)X_{p+j}^{i}ε^{p})\overline{V} _{e}^{j}  (62)

and

\overline{V}^{i}(R\overline{Z}(\overline{X})) = (∑_{|p|=0}^{r−1}(p_{j} + 1)X_{p+j}^{i}ε^{p})\overline{V} _{e}^{j}+

+ ( ∑_{|p|=r−1}(p_{j} + 1)X_{k}^{i}Z_{p+j}^{k}ε^{p})\overline{V} _{e}^{j}, (63)

we conclude that Z_{p}^{k} = 0 for |p| = r. Thus, Ψ = id and \overline{Φ} = Φ^{𝔸⊗ℝ(n,r)}. □

There are the natural embeddings of the bundle B^{r}M_{n} of r-frames of a manifold M_{n} into the bundle B^{r}(𝔸)T^{𝔸}M_{n} of 𝔸-smooth r-frames B^{r}(𝔸)T^{𝔸}M_{n} and of the structure group G_{n}^{r} of the bundle B^{r}M_{n} into the structure group G_{n}^{r}(𝔸) of the bundle B^{r}(𝔸)T^{𝔸}M_{n} (see (13)) defined by the correspondence ι : j_{0}^{r}f ↦ j_{0}^{r}(T^{𝔸}f), where f is a germ of diffeomorphism and T^{𝔸}f is the 𝔸-prolongation of f. Under these embeddings, the vector space T_{e}^{r−1}B^{r}ℝ^{n} becomes a subspace in T_{e}^{r−1}B^{r}(𝔸)𝔸^{n} which generates T_{e}^{r−1}B^{r}(𝔸)𝔸^{n} as an 𝔸-module, and each fundamental semivector field on B^{r}M_{n} is the restriction of the corresponding fundamental semivector field on B^{r}(𝔸)T^{𝔸}M_{n}. This implies the following proposition.

Proposition 9. i) Under the embedding (13), the structure form θ^{r} of B^{r}M_{n} coincides with the restriction to B^{r}M_{n} of the ℝ(n, r − 1)^{n}-valued part of the structure form Θ^{r} of B^{r}(𝔸)T^{𝔸}M_{n}, and the form Θ^{r} coincides with the 𝔸-prolongation of the form θ^{r}.

ii) If a local diffeomorphism \overline{Φ} : B^{r}(𝔸)T^{𝔸}M_{n} → B^{r}(𝔸)T^{𝔸}M′_{n} maps the structure form Θ^{r} of B^{r}(𝔸)T^{𝔸}M_{n} into the structure form Θ′^{r} of B^{r}(𝔸)T^{𝔸}M′_{n} and maps the subbundle B^{r}M_{n} to the subbundle B^{r}M′_{n}, then, in a neighborhood of every point \overline{X} ∈ B^{r}(𝔸)T^{𝔸}M_{n}, \overline{Φ} coincides with the 𝔸 ⊗ ℝ(n, r)-prolongation of a local diffeomorphism φ : M_{n} → M′_{n}.

Proof. The first part of statement i) follows from the fact that the fundamental semivector fields of the bundle B^{r}M_{n} are the restrictions of the fundamental semivector fields of the bundle B^{r}(𝔸)T^{𝔸}M_{n} corresponding to the elements of T_{e}^{r−1}B^{r}ℝ^{n} ⊂ T_{e}^{r−1}B^{r}(𝔸)𝔸^{n}. The second part of statement i) then follows from the 𝔸-smoothness of the form Θ^{r}.

If \overline{Φ} maps the structure form of the bundle B^{r}(𝔸)T^{𝔸}M_{n} into the structure form of the bundle B^{r}(𝔸)T^{𝔸}M′_{n} and the bundle B^{r}M_{n} to the bundle B^{r}M′_{n}, then the restriction Φ = \overline{Φ}|B^{r}M_{n} maps the structure form of B^{r}M_{n} into the structure form of B^{r}M′_{n}. In this case [21], Φ is the ℝ(n, r)-prolongation of a local diffeomorphism φ : M_{n} → M′_{n}, and the 𝔸⊗ℝ(n, r)-diffeomorphism \overline{Φ} coincides with the 𝔸⊗ℝ(n, r)-prolongation of φ : M_{n} → M′_{n}. □

3.8. Structure equations of the bundle B^{r}(𝔸)T^{𝔸}M_{n}. We define the T_{e}^{r−2}B^{r}(𝔸)T^{𝔸}ℝ^{n}-valued 1-form Θ̃^{r} on B^{r}(𝔸)T^{𝔸}M_{n} as follows: Θ̃^{r} = π_{r−2}^{r−1}∘Θ^{r}, where π_{r−2}^{r−1} : T_{e}^{r−1}B^{r}(𝔸)T^{𝔸}ℝ^{n} → T_{e}^{r−2}B^{r}(𝔸)T^{𝔸}ℝ^{n} is the canonical projection. Equivalently, Θ̃^{r} can be defined as the inverse image of the structure form Θ^{r−1} of B^{r−1}(𝔸)T^{𝔸}M_{n}. Let Θ̃^{r} = Θ̃^{j}e_{j}^{(r−2)} be the expansion of Θ̃^{r} in terms of the standard basis in the 𝔸⊗ℝ(n, r−2)-module T_{e}^{r−2}B^{r}(𝔸)T^{𝔸}ℝ^{n} ≡ (𝔸 ⊗ ℝ(n, r − 2))^{n}, and let Θ^{r} = Θ^{j}e_{j}^{(r−1)} be the expansion of Θ^{r} in terms of the standard basis in the 𝔸⊗ℝ(n, r−1)-module T_{e}^{r−1}B^{r}(𝔸)T^{𝔸}ℝ^{n} ≡ (𝔸 ⊗ ℝ(n, r − 1))^{n}. For two arbitrary fundamental semivector fields \overline{V}^{(r−1)} and \overline{W}^{(r−1)}, using the well-known formula for the exterior differential of a 1-form ([11], p. 36), we have

dΘ̃^{r}(\overline{V}^{(r−1)}, \overline{W}^{(r−1)}) =

= \frac{1}{2}(\overline{V}^{(r−1)}(Θ̃^{r}(\overline{W}^{(r−1)})) − \overline{W}^{(r−1)}(Θ̃^{r}(\overline{V}^{(r−1)})) − Θ̃^{r}([\overline{V}^{(r−1)}, \overline{W}^{(r−1)}])) =

= −\frac{1}{2}Θ̃^{r}([\overline{V}^{(r−1)}, \overline{W}^{(r−1)}]) = −\frac{1}{2}([\overline{V} _{e}^{(r−1)}, \overline{W} _{e}^{(r−1)}]) =

= −\frac{1}{2}(\overline{W} _{e}^{j}∂_{j}\overline{V} _{e}^{i} − \overline{V} _{e}^{j}∂_{j}\overline{W} _{e}^{i})∂_{i} =

= −\frac{1}{2}(Θ^{j}(\overline{W}^{(r−1)})∂_{j}(Θ^{i}(\overline{V}^{(r−1)})) − Θ^{j}(\overline{V}^{(r−1)})∂_{j}(Θ^{i}(\overline{W}^{(r−1)}))) =

= (Θ^{j} ∧ ∂_{j}Θ^{i})(\overline{V}^{(r−1)}, \overline{W}^{(r−1)}).

As a result of the above calculations, we obtain the following proposition.

Proposition 10. On the bundle B^{r}(𝔸)T^{𝔸}M_{n}, the following structure equations hold:

dΘ̃^{i} = Θ^{j} ∧ ∂_{j} ∘ Θ^{i}.  (64)

Note 5. For each r, one can take the inverse image Θ̃′^{r} of the form Θ̃^{r} with respect to the projection π_{r}^{∞} : B^{∞}(𝔸)T^{𝔸}M_{n} → B^{r}(𝔸)T^{𝔸}M_{n}. Since Θ̃′^{r} = π_{r−2}^{r−1} ∘ Θ̃′^{r+1}, where π_{r−2}^{r−1} : T_{e}^{r−2}B^{r}(𝔸)T^{𝔸}ℝ^{n} → T_{e}^{r−2}B^{r}(𝔸)T^{𝔸}ℝ^{n}, in the projective limit, one obtains [1] the 𝔸(n, ∞)^{n}-valued form Θ^{∞} on B^{∞}(𝔸)T^{𝔸}M_{n}. For r tending to infinity, the series of equations (64) gives the following expression for the exterior differential of the form Θ^{∞}:

d(Θ^{∞})^{i} = (Θ^{∞})^{j} ∧ ∂_{j} ∘ (Θ^{∞})^{i},  (65)

where ∂_{j} is the 𝔸-linear derivation of the algebra 𝔸(n, ∞) defined by ∂_{j}(ε^{i}) = δ_{j}^{i}.

For 𝔸 = ℝ, the equations (103) coincide with the infinite series of G.F. Laptev structure equations [18], [41], [42].

In a form similar to (103) one can represent the Maurer–Cartan equations for the Lie group G_{n}^{r}(𝔸) (see [31]).

4. Weil functors and product preserving functors on the category Mf_{tr}^{m}.

4.1. The category of m-parameter-dependent manifolds Mf_{tr}^{m}. The category Mf_{tr}^{m} is defined as follows. The objects of Mf_{tr}^{m} are the trivial fiber bundles p : M_{n}×U → U, where M_{n} is a smooth manifold and U is an open subset of ℝ^{m}. The morphisms of Mf_{tr}^{m} are the commutative diagrams of the form

\xymatrix{M × U \ar[d] \ar[r]^{f} & M′ × U′ \ar[d] \\ U \ar[r]^{_{tr_{t_{0}}}} & U′}  (66)

where f is a smooth mapping and tr_{t_{0}} : U ∋ {t^{a}} ↦ {t^{a} + t_{0}^{a}} ∈ U′ is the restriction of a translation tr_{t_{0}} : ℝ^{m} → ℝ^{m} which embeds U into U′. We will denote a morphism (66) by (f, tr_{t_{0}}) or simply by f. In terms of local coordinates (x^{i}, t^{a}) on M_{n} ×U and (x^{i′}, t^{a′}) on M′_{k} ×U′, a morphism (f, tr_{t_{0}}) is given by equations x^{i′} = f^{i′}(x^{i}, t^{a}), t^{a′} = t^{a} + t_{0}^{a}.

The category Mf^{m} is a subcategory of Mf_{tr}^{m}.

We also consider the category FM_{tr}^{m} of m-parameter-dependent fibered manifolds from Mf_{tr}^{m} whose objects are the commutative diagrams

\xymatrix{E × U \ar[r]^{p_{E}} \ar[d]_{π} & U \ar[d]^{id} \\ M_{n} × U \ar[r]^{p} & U}  (67)

where p_{E} : E × U → U and p : M_{n} × U → U are objects of Mf_{tr}^{m}, and whose morphisms are the commutative diagrams

[figure: E × U; π; f; E′ × U′; π′; M_{n} × U; p; \overline{f}; M′_{k} × U′; p′; U; id; _{tr_{t_{0}}}; U′; id; U; _{tr_{t_{0}}}; U′]  (68)

The base functor B : FM_{tr}^{m} → Mf_{tr}^{m} and the erasing functor ε : FM_{tr}^{m} → Mf_{tr}^{m} are defined as in the case of the category Mf^{m}.

The pair (φ × tr_{t_{0}}, tr_{t_{0}}), where φ × tr_{t_{0}} is the product of a smooth mapping φ : M → M′ and a translation tr_{t_{0}} : U → U′, is a morphism of Mf_{tr}^{m}. For brevity, in what follows we will denote such a morphism simply by φ × tr_{t_{0}}. The objects of the category Mf_{tr}^{m} together with all morphisms of the form φ × tr_{t_{0}} constitute a subcategory of Mf_{tr}^{m}, which will be denoted by (Mf × ℝ^{m})_{tr}. The category Mf × ℝ^{m} is a subcategory of (Mf × ℝ^{m})_{tr}. Obviously, every (Mf × ℝ^{m})_{tr}-morphism φ × tr_{t_{0}} : M_{n} × U → M′_{k} × U′ is the restriction of a morphism φ × tr_{t_{0}} : M_{n} × ℝ^{m} → M′_{k} × ℝ^{m}. More precisely, the following commutative diagram holds:

\xymatrix{M_{n} × U \ar[d]_{φ×tr_{t_{0}}} \ar[r]^{id_{M} ×i_{U}} & M_{n} × ℝ^{m} \ar[d]^{φ×tr_{t_{0}}} \\ M′_{k} × U′ \ar[r]^{id_{M′} ×i_{U′}} & M′_{k} × ℝ^{m}}  (69)

where i_{U} : U → ℝ^{m} and i_{U′} : U′ → ℝ^{m} are inclusions. Note that an (Mf ×ℝ^{m})_{tr}-morphism φ×tr_{t_{0}} : M_{n}×ℝ^{m} → M′_{k}×ℝ^{m} can be represented as the compositions (φ×id)∘(id_{M} × tr_{t_{0}}) = (id_{M′} × tr_{t_{0}})∘(φ×id), where φ × id is an Mf × ℝ^{m}-morphism.

A covariant functor F : (Mf × ℝ^{m})_{tr} → FM_{tr}^{m} satisfying the prolongation condition B ∘ F = id_{Mf×ℝ^{m}} is called a prolongation functor.

To a prolongation functor (10), one can associate the functor F_{0} : Mf → Mf defined by

F_{0} = R_{0} ∘ ε ∘ F ∘ R_{pt} .

A prolongation functor F : (Mf × ℝ^{m})_{tr} → FM_{tr}^{m} will be called a bundle functor if it satisfies the following conditions:

i) for any open subsets U′ ⊂ U ⊂ ℝ^{m} and any smooth manifold M_{n}, F (M_{n} × U′) is the restriction of F (M_{n} × U);

ii) the restriction of F to the category Mf^{m} is a bundle functor.

4.2. Products in the category Mf_{tr}^{m}. The product of two objects C_{1} = p : M_{n} × U → U and C_{2} = p′ : M′_{k} × U′ → U′ of the category Mf_{tr}^{m} in the sense of diagram (11) cannot exist because, first, the objects C_{1} and C_{2} may have distinct domains U and U′, second, the translation components tr_{t_{0}} and tr_{t′_{0}} of morphisms (f_{1}, tr_{t_{0}}) : D → C_{1} and (f_{2}, tr_{t′_{0}}) : D → C_{2} may also not coincide. For this reason, we define the product in the category Mf_{tr}^{m} only for objects p : M_{n}×U → U and p′ : M′_{k} ×U → U with the same domain U.

By the product of objects p : M × U → U and p′ : M′ × U → U in the category Mf_{tr}^{m} we will understand a triple (p′′ : M′′ × U → U, Pr, Pr′), where Pr and Pr′ are morphisms in Mf_{tr}^{m} satisfying the condition that, for any two morphisms (f, tr_{t_{0}}) : W × V → M × U and (f′, tr_{t_{0}}) : W × V → M′ × U with the same translation component tr_{t_{0}}, there exists a unique morphism (f′′, tr_{t_{0}}) : W × V → M′′ × U such that the following diagram commutes:

\xymatrix{M × U & M′′ × U \ar[l]_{Pr} \ar[r]^{^{Pr′}} & M′ × U \\  & W × V \ar[ul]^{(f, tr_{t_{0}})} \ar[u]_{(f′′, tr_{t_{0}})} \ar[ur]_{(f′, tr_{t_{0}})}}  (70)

Obviously, the triple ((M×M′)×U, pr × id_{U}, pr′ × id_{U}), where pr : M× M′ → M and pr′ : M × M′ → M′ are the projections of the product M × M′ of two manifolds in the category Mf of smooth manifolds, satisfies the above definition.

By the product of two objects E×U → M×U and E′×U → M′×U (for brevity, we will omit the projections to U in diagrams of the form (67)) of the category FM_{tr}^{m} we will understand, as in the case of the category Mf_{tr}^{m}, an object E′′ × U → M′′ × U such that, for any two morphisms from Q×V → W ×V to E ×U → M ×U and E′ ×U → M′ ×U with the same translation component tr_{t_{0}}, there exists a unique morphism from Q × V → W × V to E′′ × U → M′′ × U for which the commutative diagram

\xymatrix{E × U & E′′ × U \ar[l]_{Pr} \ar[r]^{^{Pr′}} & E′ × U \\  & Q × V \ar[ul]^{(f, tr_{t_{0}})} \ar[u]_{(f′′, tr_{t_{0}})} \ar@{-->}[ur]_{(f′, tr_{t_{0}})}}  (71)

over the commutative diagram (70) holds.

Obviously, the object (E × E′) × U → (M × M′) × U with the corresponding projections to E × U → M × U and E′ × U → M′ × U satisfy the above definition.

4.3. m-parameter families of Weil functors. By a smooth m-pa-rameter family of algebras we will mean a vector bundle 𝕍 × ℝ^{m} → ℝ^{m} with a smooth fiberwise bilinear multiplication operation ∗ : 𝕍 × 𝕍 × ℝ^{m} → 𝕍 × ℝ^{m} which is a morphism of the category Mf^{m}. Suppose that ∗ turns each fiber 𝕍_{t} = 𝕍, t ∈ ℝ^{m}, into a local Weil algebra 𝔸(t) such that the unity 1_{t} smoothly depends on t and the bundle 𝕍 × ℝ^{m} → ℝ^{m} can be represented as the Whitney sum

𝕍 × ℝ^{m} = (ℝ × ℝ^{m}) ⊕_{ℝ^{m}} (𝕍^{∘} × ℝ^{m}),  (72)

where ℝ × ℝ^{m} → ℝ^{m} is the m-parameter family of algebras of real numbers spanned by the unities 1_{t} ∈ 𝕍_{t}, t ∈ ℝ^{m}, and 𝕍^{∘}×ℝ^{m} → ℝ^{m} is the m-parameter family of nilpotent algebras whose fiber 𝕍^{∘}(t) coincides with the maximal ideal of 𝔸(t), then such a family 𝔸(t) will be called a family of Weil algebras. We will denote an m-parameter family of Weil algebras by 𝔸(t) × ℝ^{m} or simply by 𝔸(t). In a similar manner one can define a family of 𝔸(t)-modules 𝔸(t)^{n} × ℝ^{m}, where 𝔸(t)^{n} = \underbrace{𝔸(t) × ⋯ × 𝔸(t)}_{n}.

To every m-parameter family of Weil algebras 𝔸(t) × ℝ^{m} one can associate a covariant functor T̃^{𝔸(t)} : Mf × ℝ^{m} → FM^{m} called an m-pa-rameter family of Weil functors [3]. There are several descriptions of the Weil functor (see, e. g., [14]). Within the framework of the problem studied in this section, it is convenient to consider the construction of Weil functor based on the use of local charts. Define the action of T̃^{𝔸(t)} on ℝ × ℝ^{m} as follows: T̃^{𝔸(t)}(ℝ × ℝ^{m}) = 𝔸(t) × ℝ^{m}. For an Mf × ℝ^{m}-mor-phism f × id : ℝ × ℝ^{m} → ℝ × ℝ^{m}, the morphism T̃^{𝔸(t)}(f × id) : 𝔸(t) × ℝ^{m} → 𝔸(t) × ℝ^{m}, (x + X^{∘} , t) ↦ (y + Y^{∘} , t) is defined by the equations

y + Y^{∘} = f(x) + ∑_{p=1}^{∞} \frac{1}{p!} \frac{d^{p}f(x)}{dx^{p}} X^{∘}^{p},

where the coordinates (x + X^{∘}, t) correspond to the decomposition (72) and the summation, for each t ∈ ℝ^{m}, is over a finite number of summands depending on the height of 𝔸(t).

We let T̃^{𝔸(t)}(ℝ^{n} × ℝ^{m}) = 𝔸(t)^{n} × ℝ^{m} and define T̃^{𝔸(t)}(f × id) : 𝔸(t)^{n} × ℝ^{m} → 𝔸(t)^{k} × ℝ^{m} by

y^{j} + Y^{∘}^{j} = f^{j}(x^{1}, …, x^{n}) + ∑_{|p|≥1}^{∞} \frac{1}{p!} \frac{∂^{|p|}f^{j}(x^{1}, …, x^{n})}{∂x^{p}} X^{∘}^{p}, (73)

where p = (p_{1}, …, p_{n}) is a multiindex.

For an open subset U ⊂ ℝ^{n}, we let T̃^{𝔸(t)}(U × ℝ^{m}) = (π_{𝔸(t)}^{n})^{−1}(U × ℝ^{m}), where π_{𝔸(t)} : 𝔸(t) × ℝ^{m} → ℝ × ℝ^{m} is the fiberwise epimorphism of algebras, and define the action of T̃^{𝔸(t)} on morphisms f × id : U × ℝ^{m} → U × ℝ^{m} by the same relations (73).

Let now M_{n} be an arbitrary n-dimensional smooth manifold with atlas consisting of charts (U_{α}, h_{α}), α ∈ A, with transition functions h_{αβ} = h_{α} ∘ h_{β}^{−1} : h_{β}(U_{αβ}) → h_{α}(U_{αβ}). Then the collection {(U_{α} × ℝ^{m}, h_{α} × id)}_{α∈A} is an atlas on M_{n} ×ℝ^{m}. The collection of FM^{m}-morphisms {T̃^{𝔸(t)}(h_{αβ} × id)} allows ones to glue the domains {T̃^{𝔸(t)}(U_{α} ×ℝ^{m})}_{α∈A} and obtain the total space of the bundle T̃^{𝔸(t)}(M_{n} ×ℝ^{m}) → M_{n} ×ℝ^{m}. Let f : M_{n} → M′_{k} be a smooth mapping of an n-dimensional manifold M_{n} to a k-dimen-sional manifold M′_{k}. Then we define the morphism T̃^{𝔸(t)}(f × id) as the mapping given locally by equations (73). We have T̃^{𝔸(t)}(f ∘g) = T̃^{𝔸(t)}(f)∘ T̃^{𝔸(t)}(g) and T̃^{𝔸(t)}(id) = id. Therefore, T̃^{𝔸(t)} is a covariant functor from the category Mf × ℝ^{m} to the category FM^{m}.

The functor T̃^{𝔸(t)} : Mf × ℝ^{m} → FM^{m} constructed above is called an m-parameter family of Weil functors.

In the case when m = 1 and the algebra 𝔸(t) does not depend on t, the bundle T̃^{𝔸(t)}(M_{n} × ℝ) turns out to be a time-dependent Weil bundle T^{𝔸}M_{n} × ℝ studied by M. Doupovec and I. Kolář [4].

4.4. Product preserving bundle functors on (Mf × ℝ^{m})_{tr}. The restriction of a product preserving bundle functor F : Mf_{tr}^{m} → FM_{tr}^{m} to the subcategory (Mf × ℝ^{m})_{tr} is a product preserving bundle functor \overline{F} : (Mf ×ℝ^{m})_{tr} → FM_{tr}^{m}. For this reason, we first consider an arbitrary product preserving bundle functor G : (Mf × ℝ^{m})_{tr} → FM_{tr}^{m}. By (69), a morphism φ×tr_{t_{0}} : M_{n} ×U → M′_{k} ×U′ is the restriction of a morphism φ × tr_{t_{0}} : M_{n} × ℝ^{m} → M′_{k} × ℝ^{m}, which, as was mentioned above, can be represented in the form (φ × id) ∘ (id_{M} × tr_{t_{0}}) = (id_{M′} × tr_{t_{0}}) ∘ (φ × id), where φ × id is a morphism of the category Mf × ℝ^{m}. Then we have the following commutative diagram:

\xymatrix{M_{n} × ℝ^{m} \ar[d]_{id_{Mn} × tr_{t_{0}}} \ar[r]^{φ×id} & M′_{k} × ℝ^{m} \ar[d]^{id_{M′_{k}} × tr_{t_{0}}} \\ M_{n} × ℝ^{m} \ar[r]^{φ×id} & M′_{k} × ℝ^{m}}  (74)

Applying the functor G to diagram (74), we obtain the commutative diagram

\xymatrix{G(M_{n} × ℝ^{m}) \ar[d]_{G(id_{Mn} × tr_{t_{0}})} \ar[r]^{G(φ×id)} & G(M′_{k} × ℝ^{m}) \ar[d]^{G(id_{M′_{k}} × tr_{t_{0}})} \\ G(M_{n} × ℝ^{m}) \ar[r]^{G(φ×id)} & G(M′_{k} × ℝ^{m})}  (75)

The horizontal arrows of diagram (74) are morphisms of the category Mf × ℝ^{m}. Therefore, the restriction of G to the category Mf × ℝ^{m} coincides (up to an equivalence) with some m-parameter family of Weil functors T̃^{𝔸(t)} : Mf × ℝ^{m} → FM^{m}. Let 𝔸 = 𝔸(0). Passing to the restriction of the upper arrow of diagram (75) to 0 ∈ ℝ^{m} (applying the functor R_{0}) and the restriction of the lower arrow to t_{0} ∈ ℝ^{m}, we obtain a commutative diagram which establishes the natural equivalence of the Weil functors T^{𝔸} → T^{𝔸(t_{0})}. The diagram obtained in the same manner for the morphism φ × tr_{−t_{0}} gives the inverse natural equivalence.

Consider now the family μ = {G(id_{M_{n}} × tr_{−t})|_{t}, t ∈ ℝ^{m}} of natural equivalences. It depends smoothly on t ∈ ℝ^{m}. These equivalences and the projections pr_{1} = μ and pr_{2} = \overline{F} (p) introduce on G(M_{n} × ℝ^{m}) the structure of the product T^{𝔸}M_{n} × ℝ^{m} such that G(φ × id) = T^{𝔸}φ × id and G(id_{M_{n}} × tr_{t_{0}}) = id_{T^{𝔸}M_{n}} × tr_{t_{0}}. Then G(φ × tr_{t_{0}}) = T^{𝔸}φ × tr_{t_{0}} (with respect to the above introduced product structure) for every morphism φ × tr_{t_{0}} ∈ Mor((Mf × ℝ^{m})_{tr}). For a local algebra 𝔸, denote by T^{𝔸} : (Mf × ℝ^{m})_{tr} → (FM × ℝ^{m})_{tr} the functor whose action on objects and morphisms of the category (Mf × ℝ^{m})_{tr} is defined, respectively, as follows:

p : M_{n} × U → U ⟼ \xymatrix{T^{𝔸}M_{n} × U \ar[d]_{π×id} \ar[r]^{p^{𝔸}} & U \ar[d]^{id} \\ M_{n} × U \ar[r]^{p} & U}  (76)

\xymatrix{M_{n} × U \ar[d] \ar[r]^{φ×tr_{t_{0}}} & M′_{k} × U′ \ar[d] \\ U \ar[r]^{_{tr_{t_{0}}}} & U′} ⟼ \xymatrix{T^{𝔸}M_{n} × U \ar[d]_{π×id} \ar[r]^{T^{𝔸}φ×tr_{t_{0}}} & T^{𝔸}M′_{k} × U′ \ar[d]^{π′×id} \\ M_{n} × U \ar[r]^{φ×tr_{t_{0}}} & M′_{k} × U′} (77)

As a result of the above discussion, we obtain the following theorem.

Theorem 3. A product preserving bundle functor G : (Mf × ℝ^{m})_{tr} → FM_{tr}^{m} is naturally equivalent to the functor T^{𝔸} : (Mf ×ℝ^{m})_{tr} → (FM× ℝ^{m})_{tr} determined by some Weil algebra 𝔸.

4.5. The generalized Weil functor T̂_{σ}^{𝔸}. Applying the Weil functor T^{𝔸} defined by a Weil algebra 𝔸 to an object p : M_{n} × U → U of the category Mf_{tr}^{m}, we obtain the bundle T^{𝔸}p : T^{𝔸}(M_{n} × U) → T^{𝔸}U. Let

σ : U → T^{𝔸}U ≡ U × 𝔸^{∘}^{m} be a section {t^{a}} ↦ ({t^{a}}, {σ^{∘}^{a}}), a = 1, …, m,

given by m constant elements σ^{∘}^{a} of 𝔸^{∘}. Denote by T̂_{σ}^{𝔸}(M × U) → U the pullback of the bundle T^{𝔸}p : T^{𝔸}(M_{n} × U) → T^{𝔸}U under the mapping σ : U → T^{𝔸}U. We have the following commutative diagram:

[figure: M × U; id; M × U; T̂_{σ}^{𝔸}(M × U); π_{M}; T̂_{σ}^{𝔸}(p); σ_{M}^{∗}; T^{𝔸}(M × U); π_{M}; T^{𝔸}(p); U; id; U; U; id; σ; T^{𝔸}U; π_{pt}]  (78)

The left square of diagram (78)

\xymatrix{T̂_{σ}^{𝔸}(M × U) \ar[d] \ar[r] & M × U \ar[d] \\ U \ar[r] & U}  (79)

is an object of the category FM_{tr}^{m}. Applying the pullback construction to the 𝔸-prolongation

\xymatrix{T^{𝔸}(M × U) \ar[d] \ar[r]^{T^{𝔸}f} & T^{𝔸}(M′ × U′) \ar[d] \\ T^{𝔸}U \ar[r]^{T^{𝔸} tr_{t_{0}}} & T^{𝔸}U′}  (80)

of a morphism

\xymatrix{M × U \ar[d] \ar[r]^{f} & M′ × U′ \ar[d] \\ U \ar[r]^{_{tr_{t_{0}}}} & U′}  (81)

we obtain the diagram

[figure: M × U; f; M′ × U′; T̂_{σ}^{𝔸}(M × U); π_{M}; T̂_{σ}^{𝔸}(f); T̂_{σ}^{𝔸}(M′ × U′); π_{M′}; U; _{tr_{t_{0}}}; U′; U; id; _{tr_{t_{0}}}; U′; id]  (82)

which is a morphism in FM_{tr}^{m}. Obviously, for the composition f ∘ g of morphisms g and f of the category we have T̂_{σ}^{𝔸}(f ∘ g) = T̂_{σ}^{𝔸}(f) ∘ T̂_{σ}^{𝔸}(g) and T̂_{σ}^{𝔸}(id) = id.

Thus, the correspondence T̂_{σ}^{𝔸} which assigns to an object p : M_{n} × U → U of Mf_{tr}^{m} the object (79) of FM^{m} and to a morphism (81) the morphism (82) is a functor T̂_{σ}^{𝔸} : Mf_{tr}^{m} → FM^{m}, which will be called a generalized Weil functor. Obviously, T̂_{σ}^{𝔸} : Mf_{tr}^{m} → FM^{m} preserves products.

If, in terms of local coordinates, a morphism (81) is given by equations y^{i′} = f^{i′}(x^{j}, t^{a}), t^{a′} = t^{a} + t_{0}^{a}, then the morphism (80), in terms of the induces 𝔸-coordinates {X^{j}, S^{a} = t^{a} +S^{∘}^{a}} on T^{𝔸}(M_{n} ×U) and {Y^{i′}, S^{a′} = t^{a′} + S^{∘}^{a′}} on T^{𝔸}(M′_{k} × U′) is of the form (see [33]):

Y^{i′} = f^{i′}(x, t) + ∑_{p+s=1}^{q} \frac{1}{p!s!} \frac{∂^{|p+s|}f^{i′}}{∂x^{p}∂t^{s}} X^{∘}^{p} S^{∘}^{s}, S^{a′} = S^{a} + t_{0}^{a},

whence it follows that, in terms of the induced coordinates {X^{j}, t^{a}} on T̂_{σ}^{𝔸}(M_{n} × U) and {Y^{i′}, t^{a′}} on T̂_{σ}^{𝔸}(M′_{k} × U′), the morphism (82) is of the form

Y^{i′} = f^{i′}(x, t) + ∑_{p+s=1}^{q} \frac{1}{p!s!} \frac{∂^{|p+s|}f^{i′}}{∂x^{p}∂t^{s}} X^{∘}^{p} σ^{∘}^{s}.  (83)

Equations (83) can be rewritten in the form

Y^{i′} = f^{i′}(x, t) + ∑_{p=0}^{q} \frac{1}{p!} \frac{∂^{|p|}}{∂x^{p}} \{∑_{s=1}^{q} \frac{1}{s!} \frac{∂^{|s|}f^{i′}}{∂t^{s}} σ^{∘}^{s}\} X^{∘}^{p} =

= f̂^{i}(x, t) + ∑_{p=1}^{q} \frac{1}{p!} \frac{∂^{|p|}f̂^{i′}}{∂x^{p}} X^{∘}^{p},

where

f̂^{i′}(x, t) = f^{i′}(x, t) + ∑_{s=1}^{q} \frac{1}{s!} \frac{∂^{|s|}f^{i′}}{∂t^{s}} σ^{∘}^{s},

whence it follows that the restriction of the morphism T̂_{σ}^{𝔸}(f, tr_{t_{0}}) to each fiber over U is an 𝔸-smooth mapping.

In what follows it will be convenient to use the following explicit construction for the bundle T̂_{σ}^{𝔸}(M_{n} × U). An element of T̂_{σ}^{𝔸}(M_{n} × U) can be considered as the 𝔸-velocity j^{𝔸}g of a germ g : (ℝ^{ℓ}, 0) → M_{n} × U such that the following diagram is commutative:

\xymatrix{(ℝ^{ℓ}, 0) \ar[r]^{g} \ar[d]_{id} & M_{n} × U \ar[d]^{id} \\ (ℝ^{ℓ}, 0) \ar[r]^{tr_{t_{0}} ∘ σ̂} & U}  (84)

where σ̂ : (ℝ^{ℓ}, 0) → (ℝ^{m}, 0) is a germ with j^{𝔸}σ̂ = {σ^{∘}^{a}}, a = 1, …, m. Then the morphism T̂_{σ}^{𝔸}(f) defined by (82) can be written in the form

T̂_{σ}^{𝔸}(f) : j^{𝔸}g ↦ j^{𝔸}(f ∘ g)  (85)

4.6. Product preserving bundle functors on the category Mf_{tr}^{m}. As in the case of the category Mf^{m}, we introduce the functor Φ : Mf_{tr}^{m} → (Mf ×ℝ^{m})_{tr} which assigns to a bundle M_{n}×U → U the bundle (M_{n} × U) × U → U and to a morphism (f, tr_{t_{0}}) : M_{n} × U → M′_{k} × U′, (x, t) ↦ (f(x, t), t+t_{0}), the morphism Φ(f, tr_{t_{0}}) : (M_{n} ×U)×U → (M′_{k} × U′) × U′, ((x, s), t) ↦ ((f(x, s), s + t_{0}), t + t_{0}). The sections σ_{M} : M_{n} × ℝ^{m} → (M_{n} × ℝ^{m}) × ℝ^{m}, σ_{M}(x, t) = ((x, t), t), define a natural transformation of functors σ : Id_{Mf_{tr}^{m}} → Φ. In fact, we have the commutative diagram

\xymatrix{M_{n} × U \ar[r]^{f} \ar[d]_{σ_{M}} & M′_{k} × U′ \ar[d]^{σ_{M′}} \\ (M_{n} × U) × U \ar[r]^{Φ(f)} & (M′_{k} × U′) × U′}  (86)

\xymatrix{(x, t) \ar[r]^{f} \ar[d]_{σ_{M}} & (y, t + t_{0}) \ar[d]^{σ_{M′}} \\ ((x, t), t) \ar[r]^{Φ(f)} & ((y, t + t_{0}), t + t_{0}))}

Let now F : Mf_{tr}^{m} → FM_{tr}^{m} be a product preserving bundle functor, and let \overline{F} : (Mf × ℝ^{m})_{tr} → FM_{tr}^{m} be the restriction of F to the subcategory (Mf × ℝ^{m})_{tr}. Applying F to diagram (86), we obtain the diagram

\xymatrix{F (M_{n} × U) \ar[r]^{F (f)} \ar[d]_{F (σ_{M})} & F (M′_{k} × U′) \ar[d]^{F (σ_{M′})} \\ F ((M_{n} × U) × U) \ar[r]^{F Φ(f)} & F ((M′_{k} × U′) × U′)}  (87)

\xymatrix{(X, t) \ar[r]^{F (f)} \ar[d]_{F (σ_{M})} & (Y, t + t_{0}) \ar[d]^{F (σ_{W})} \\ ((X, S), t) \ar[r]^{F Φ(f)} & ((Y, S + t_{0}), t + t_{0})}

whose lower arrow is the morphism \overline{F} Φ(f, tr_{t_{0}}).

In accordance with diagram (70), the section σ_{M} is the product of the two morphisms id_{M_{n}×ℝ^{m}} and σ′_{M} : M_{n} × ℝ^{m} → ℝ^{m} × ℝ^{m}, σ′_{M} : (x, t) = (t, t). Thus, we have the commutative diagram

\xymatrix{M_{n} × ℝ^{m} & (M_{n} × ℝ^{m}) × ℝ^{m} \ar[l]_{Pr} \ar[r]^{^{Pr′}} & ℝ^{m} × ℝ^{m} \\  & M_{n} × ℝ^{m} \ar@{-->}[ul]^{id} \ar[u]_{σ_{M}} \ar@{-->}[ur]_{σ′_{M}}}  (88)

Applying to diagram (88) the functor F, we obtain the diagram

\xymatrix{F (M_{n} × ℝ^{m}) & F ((M_{n} × ℝ^{m}) × ℝ^{m}) \ar[l]_{Pr} \ar[r]^{^{Pr′}} & F (ℝ^{m} × ℝ^{m}) \\  & F (M_{n} × ℝ^{m}) \ar@{-->}[ul]^{id} \ar[u]_{F (σ_{M})} \ar@{-->}[ur]_{F (σ′_{M})}} (89)

Hence the morphism F (σ_{M}) is determined by F (σ′_{M}). Applying F to the commutative diagram

\xymatrix{M_{n} × ℝ^{m} \ar[rr]^{pt} \ar[dr]_{σ′_{M}} &  & pt ×ℝ^{m} \ar@{-->}[dl]^{σ′_{pt}} \\  & ℝ^{m} × ℝ^{m}}

we conclude that F (σ′_{M}) is completely determined by the morphism F (σ′_{pt}) : pt ×ℝ^{m} → T^{𝔸}ℝ^{m}×ℝ^{m}, which is in fact a section ℝ^{m} → 𝔸^{m}×ℝ^{m}

of the form F (σ′_{pt}) : {t^{a}} ↦ ({S^{a}(t)}, {t^{a}}), S^{a}(t) = t^{a} +σ^{∘}^{a}(t), σ^{∘}^{a}(t) ∈ 𝔸^{∘},

a = 1, …, m. From diagram (87), we have t+t_{0} +σ^{∘}(t) = t+t_{0} +σ^{∘}(t+t_{0}). Therefore, σ^{∘}^{a}(t) = σ^{∘}^{a}(t+t_{0}). Consequently, the 𝔸^{∘}-valued functions σ^{∘}^{a}(t) are constants.

The lower arrow of diagram (87) is an (FM × ℝ^{m})_{tr}-morphism T^{𝔸}f × tr_{t_{0}}. In terms of local coordinates, it is of the form [33]:

Y^{i′} = f^{i′}(x, t) + ∑_{p+s=1}^{q} \frac{1}{p!s!} \frac{∂^{|p+s|}f^{i′}}{∂x^{p}∂t^{s}} X^{∘}^{p} S^{∘}^{s}.

Then the morphism F (f, tr_{t_{0}}) (the upper arrow of diagram (87)), in terms of local coordinates, is is given by the equations

Y^{i′} = f^{i′}(x, t) + ∑_{p+s=1}^{q} \frac{1}{p!s!} \frac{∂^{|p+s|}f^{i′}}{∂x^{p}∂t^{s}} X^{∘}^{p} σ^{∘}^{s},

which coincide with (83). Thus, the action of F on morphisms is completely determined by the collection of elements σ^{∘}^{a} ∈ 𝔸^{∘}, a = 1, …, m, which defines the section F (σ′_{pt}).

As a result of the above discussion, we obtain the following theorem.

Theorem 4. A product preserving bundle functor F : Mf_{tr}^{m} → FM_{tr}^{m} is naturally equivalent to a generalized Weil functor T̂_{σ}^{𝔸} : Mf_{tr}^{m} → FM_{tr}^{m} defined by some Weil algebra 𝔸 and a collection of elements σ^{∘}^{a} ∈ 𝔸^{∘}, a = 1, …, m.

5. Higher order geometry of manifolds from Mf_{tr}^{m}.

5.1. Higher order frame bundles of manifolds from Mf_{tr}^{m}. Denote by T̂_{pr}^{ℝ(n+m,r)}(M′_{k} × U) the bundle T̂_{σ}^{ℝ(n+m,r)}(M′_{k} × U) corresponding to the section σ defined by σ̂ = pr : (ℝ^{n} × ℝ^{m}, 0) → (ℝ^{m}, 0) (see (84)), i. e., by σ^{∘}^{a} = ν^{a}, where {ν^{a}}, a = 1, …, m, is the standard pseudobasis in ℝ(m, r). Let B̂^{r}(M_{n} × U) be the set of all r-jets of invertible germs of morphisms (f, tr_{t_{0}}) : (ℝ^{n} × ℝ^{m}, (0, 0)) → (M_{n} × ℝ^{m}, (x, t)), (u^{j}, t^{a}) ↦ (x^{i}(u^{j}, t^{a}), t^{a′} = t^{a} + t_{0}^{a}), from the category Mf_{tr}^{m}. The set B̂^{r}(M_{n} × U) is an open subset in the bundle T̂_{pr}^{ℝ(n+m,r)}(M_{n} × U), and so it inherits from T̂_{pr}^{ℝ(n+m,r)}(M_{n} ×U) the structure of a smooth manifold. In addition, B̂^{r}(M_{n}×U) is a locally trivial fiber bundle over M_{n}×U which is an object of the category FM_{tr}^{m}. The standard fiber of B̂^{r}(M_{n} × U) is the Lie group D̂^{r}(n, m) consisting of all r-jets of invertible germs of morphisms (f, id) : (ℝ^{n} × ℝ^{m}, (0, 0)) → (ℝ^{n} × ℝ^{m}, (0, 0)). The Lie group D̂^{r}(n, m) acts on the right on B̂^{r}(M_{n} × U) by the law of composition of jets. Therefore, B̂^{r}(M_{n} × U) is a principal bundle.

The Weil algebra ℝ(n + m, r) considered as the algebra of r-jets of germs (ℝ^{n} × ℝ^{m}, 0) → ℝ, (u^{j}, t^{a}) ↦ x = f(u^{j}, t^{a}), contains the two Weil subalgebras consisting of r-jets of germs which do not depend, respectively, on u^{j} and t^{a}. These subalgebras are isomorphic, respectively, to ℝ(n, r) and ℝ(m, r). In what follows by ℝ(n, r) and ℝ(m, r) we will mean the above mentioned subalgebras of the algebra ℝ(n + m, r). The subalgebras ℝ(n, r) and ℝ(m, r) generate the algebra ℝ(n + m, r), and ℝ(n + m, r) ≅ ℝ(n, r)ˆ⊗ℝ(m, r) = ℝ(n, r) ⊗ ℝ(m, r)/𝔪(ℝ(n, r) ⊗ ℝ(m, r))^{r+1}.

Let {ε^{i}}, i = 1, …, n, denote the standard pseudobasis in ℝ(n, r) and {ν^{a}}, a = 1, …, m, the standard pseudobasis in ℝ(m, r). The total collection {ε^{i}, ν^{a}}, i = 1, …, n, a = 1, …, m, is a pseudobasis in ℝ(n + m, r) ≅ ℝ(n, r)ˆ⊗ℝ(m, r). In terms of this pseudobasis, the local coordinates on B̂^{r}(M_{n} × U) induced by coordinates on M_{n} × U can be written in the form

X^{i} = ∑_{|p|+|s|=0}^{r} x_{ps}^{i}ε^{p}ν^{s}.  (90)

In terms of the coordinates (90), the composition Y = X ∘ Z in the Lie group D̂^{r}(n, m) and the right action Y = X ∘ Z of D̂^{r}(n, m) on B̂^{r}(M_{n} × U) are, respectively, of the form

Y^{i} = ∑_{|p|+|s|=1}^{r} x_{ps}^{i} Z^{∘}^{p}ν^{s} and Y^{i} = ∑_{|p|+|s|=0}^{r} x_{ps}^{i} Z^{∘}^{p}ν^{s}.  (91)

The principal fiber bundle (B̂^{r}(M_{n} ×U), D̂^{r}(n, m), M_{n} ×U) will be called the r-frame bundle of M_{n} × U.

Thus, to a manifold M_{n} ×U, there is naturally associated the sequence of principal bundles of higher order frames

M_{n} ⟵^{π_{0}^{1}} B̂^{1}(M_{n} × U) ⟵^{π_{1}^{2}} … ⟵^{π_{r−1}^{r}} B̂^{r}(M_{n} × U) ⟵^{π_{r}^{r+1}} …

… ⟵ B̂^{∞}(M_{n} × U) , (92)

where B̂^{∞}(M_{n} × U) is the bundle of infinite order frames of M_{n} × U, the limit of the projective system M_{n} × U ← B̂^{1}(M_{n} × U) ← … endowed with the corresponding structure of an infinite-dimensional smooth manifold in the sense of Bernshtein–Rozenfeld [1]. The bundle B̂^{∞}(M_{n} × U) is formed by the infinite order jets of germs of morphisms (f, tr_{t}) : (ℝ^{n} × ℝ^{m}, (0, 0)) → (M_{n} × ℝ^{m}, (x, t)). The group D̂^{∞}(n, m) consisting of the infinite order jets of germs of invertible morphisms (f, id) : (ℝ^{n} × ℝ^{m}, (0, 0)) → (ℝ^{n} × ℝ^{m}, (0, 0)) acts naturally on the right on B̂^{∞}(M_{n} × U).

Proposition 11. i) The Lie group D̂^{r}(n, m) is isomorphic to the Lie group of ℝ(m, r)-linear automorphisms of the algebra ℝ(n, r)ˆ⊗ℝ(m, r).

ii) The Lie algebra 𝔡^{r}(n, m) of the Lie group D̂^{r}(n, m) is isomorphic to the Lie algebra of ℝ(m, r)-linear derivations of the algebra ℝ(n, r)ˆ⊗ℝ(m, r) with bracket

[D_{1}, D_{2}] = D_{2} ∘ D_{1} − D_{1} ∘ D_{2}.  (93)

Proof. The proof of this proposition is similar to that of Proposition 1 (see also [31] for the case of algebra 𝔸(n, q) = ℝ(n, q)ˆ⊗𝔸). The algebra ℝ(n, r)ˆ⊗ℝ(m, r) can be considered as the algebra of ℝ(n + m, r)-jets of morphisms (f, id) : (ℝ^{n} × ℝ^{m}, (0, 0)) → (ℝ × ℝ^{m}, (x, 0)). Then the action of the ℝ(n, r)ˆ⊗ℝ(m, r)-jet of a germ of morphism (g, id) : (ℝ^{n} × ℝ^{m}, (0, 0)) → (ℝ^{n} × ℝ^{m}, (0, 0)) on the right on ℝ(n, r)ˆ⊗ℝ(m, r) by the law of composition of jets is an automorphism. If a germ (f, id) : (ℝ^{n} × ℝ^{m}, (0, 0)) → (ℝ × ℝ^{m}, (x, 0)) does not depend on u ∈ ℝ^{n} (i. e., is of the form x = f(t^{a})), then (f, id) ∘ (g, id) = (f, id) and, therefore, the element j^{ℝ(n,r)ˆ⊗ℝ(m,r)}g of D̂^{r}(n, m) acts on the subalgebra ℝ(m, r) as the identity transformation.

The identification of the tangent spaces to the algebra ℝ(n, r)ˆ⊗ℝ(m, r) with the same algebra converts the fundamental vector fields of the action of D̂^{r}(n, m) on ℝ(n, r)ˆ⊗ℝ(m, r) into the ℝ(m, r)-linear derivations of ℝ(n, r)ˆ⊗ℝ(m, r). □

5.2. The structure form of the bundle B̂^{r}(M_{n} × U). Let φ : (ℝ^{n} × ℝ^{m}, (0, 0)) → (M_{n} × ℝ^{m}, (x, t_{0})) be a germ of invertible morphism, and let X = j^{r}φ be the r-frame from B̂^{r}(M_{n} × U) defined by φ. The tangent mapping to the germ T̂_{pr}^{ℝ(n+m,r)}(φ) at e ∈ B̂^{r}(ℝ^{n}×ℝ^{m}) is an isomorphism of tangent spaces

T T̂_{pr}^{ℝ(n+m,r)}(φ) : T_{e}B̂^{r}(ℝ^{n} × ℝ^{m}) → T_{X} B̂^{r}(M_{n} × U).

But the r-frame X = j^{r}φ defines only the isomorphism

T^{r−1}T̂_{pr}^{ℝ(n+m,r)}(φ) : T_{e}^{r−1}B̂^{r}(ℝ^{n} × ℝ^{m}) → T_{X}^{r−1}B̂^{r}(M_{n} × U),  (94)

where T^{r−1}B̂^{r}(M_{n} × U) is the pullback of the tangent bundle T B̂^{r}(M_{n} × U) under the projection π_{r−1}^{r} : B̂^{r}(M_{n}×U) → B̂^{r−1}(M_{n}×U). An element of T_{X}^{r−1}B̂^{r}(M_{n} ×U) can be considered as a tangent vector to B̂^{r}(M_{n} ×U) given up to an addend belonging to the kernel of the projection T_{X}π_{r−1}^{r} or, in terms of the algebra ℝ(n + m, r), up to an addend belonging to the submodule

𝔪(ℝ(n + m, r))^{r}V_{X} B̂^{r}(M_{n} × U)

of the vertical tangent ℝ(n + m, r)-module V_{X} B̂^{r}(M_{n} × U) (the tangent space to the fiber t = t_{0} of the projection of B̂^{r}(M_{n} ×U) to U) generated by the r-th power of the maximal ideal 𝔪(ℝ(n + m, r)) of ℝ(n + m, r). Thus, a fiber T_{X}^{r−1}B̂^{r}(M_{n} × U) of the bundle T^{r−1}B̂^{r}(M_{n} × U) can be considered as the quotient space

T_{X} B̂^{r}(M_{n} × U)/𝔪(ℝ(n + m, r))^{r}V_{X} B̂^{r}(M_{n} × U).

Then the bundle T^{r−1}B̂^{r}(M_{n} × U) is the quotient bundle of the vector bundle T B̂^{r}(M_{n} × U) → B̂^{r}(M_{n} × U) by the subbundle

𝔪(ℝ(n + m, r))^{r}V B̂^{r}(M_{n} × U)

of the vertical tangent bundle V B̂^{r}(M_{n} × U).

Each element V_{e}^{(r−1)} of T_{e}^{r−1}B̂^{r}(ℝ^{n} × ℝ^{m}) defines, by means of the mappings (94), a section V^{(r−1)} of the bundle T^{r−1}B̂^{r}(M_{n} × U). We will call the section V^{(r−1)} the fundamental semivector field on B̂^{r}(ℝ^{n} × ℝ^{m}) corresponding to V_{e}^{(r−1)}.

The inverse mappings to (94) define the 1-form θ̂^{r} on B̂^{r}(M_{n} ×U) with values in T_{e}^{r−1}B̂^{r}(ℝ^{n} × ℝ^{m}):

θ̂_{X}^{r}(V_{X}) = (T^{r−1}T̂_{pr}^{ℝ(n+m,r)}(φ))^{−1}(T π_{r−1}^{r}(V_{X})).  (95)

The form θ̂^{r} will be called the structure form of the bundle B̂^{r}(M_{n} × U).

By a local isomorphism φ : M_{n} × U → M′_{n} × U′ we will mean a morphism of the category Mf_{tr}^{m} which is a local diffeomorphism.

Theorem 5. Let Φ : B̂^{r}(M_{n} × U) → B̂^{r}(M′_{n} × U′) be a local diffeomorphism which maps the structure form θ̂^{r} of the bundle B̂^{r}(M_{n} × U) into the structure form θ̂′^{r} of the bundle B̂^{r}(M′_{n} × U′). Then in a neighborhood of every point X ∈ B̂^{r}(M_{n} × U) the mapping Φ coincides with the ℝ(n, r)ˆ⊗ℝ(m, r)-prolongation of a local isomorphism φ : M_{n}×U → M′_{n}× U′.

Proof. The mapping Φ maps the fundamental semivector field on B̂^{r}(M_{n}× U) corresponding to an element V_{e} ∈ T_{e}^{r−1}B̂^{r}(ℝ^{n} × ℝ^{m}) into the fundamental semivector field on B̂^{r}(M′_{n} × U′) corresponding to the same element V_{e}. If the projection of V_{e} to the tangent space T_{0,0}(ℝ^{n} × ℝ^{m}) is the zero vector, then the projections of the corresponding fundamental semivector fields to M_{n} ×U and M′_{n} ×U′ are the zero vector fields. Hence it follows that the local diffeomorphism Φ projects into a local diffeomorphism φ : M_{n} ×U → M′_{n} ×U′. For the same reason, Φ and φ project into a local diffeomorphism ψ : U → U′, i. e., in terms of local coordinates, φ is of the form x^{i′} = φ^{i′}(x^{j}, t^{a}), t^{a′} = ψ^{a′}(t^{b}). In addition, from (95) and (94) it follows that the projection of a fundamental semivector field to ℝ^{m} is a constant vector field. Consequently, constant vector fields on ℝ^{m} are invariant under the mapping ψ. But then \frac{∂t^{a′}}{∂t^{b}} = δ_{b}^{a} and t^{a′} = t^{a} + t_{0}^{a}, i. e., ψ is a translation of ℝ^{m}. Thus, φ : M_{n}×U → M′_{n}×U′ is a morphism of the category Mf_{tr}^{m}.

Now, following the arguments from Section 1.3.1 of [21] (see also the proof of Theorem 2), we consider the composition

Ψ^{r} = (φ^{ℝ(n,r)ˆ⊗ℝ(m,r)})^{−1} ∘ Φ : B̂^{r}(M_{n} × U) → B̂^{r}(M_{n} × U).  (96)

The local diffeomorphism Ψ^{r} preserves the structure form and projects into the identity diffeomorphism of M_{n} × U to itself. Therefore, Ψ^{r} consists of right translations R_{Z(x,t)}, Z ∈ D̂^{r}(n, m), of the fibers (π_{0}^{r})^{−1}(x, t) of the bundle B̂^{r}(M_{n} × U). It remains to show that Ψ^{r} is the identity mapping.

For r = 1, in terms of local coordinates, the right translation R_{Z} : X ↦ Y is of the form y_{j}^{i} = x_{k}^{i}z_{j}^{k}, y_{a}^{i} = x_{k}^{i}z_{a}^{k} + x_{a}^{i}. Let v_{e} ∈ T_{e}^{0}B̂^{1}(ℝ^{n} × ℝ^{m}) be an arbitrary element, and let (v_{e}^{i}, v_{e}^{a}) be the coordinates of v_{e}. Equating the coordinates (v^{i}, v^{a}) of the values of the fundamental semivector field corresponding to v_{e} at X and Y, we obtain

x_{j}^{i}v_{e}^{j} + x_{a}^{i}v_{e}^{a} = x_{k}^{i}z_{j}^{k}v_{e}^{j} + (x_{k}^{i}z_{a}^{k} + x_{a}^{i})v_{e}^{a},

whence it follows that z_{j}^{k} = δ_{j}^{k}, z_{a}^{k} = 0, that is, Z(x, t) = e ∈ D̂^{1}(n, m).

Assuming that the statement holds for r = 1, …, r − 1, we conclude that the mapping (96) fibers over the identity mapping B̂^{r−1}(M_{n} ×U) → B̂^{r−1}(M_{n} × U). Then, as in the case r = 1, equating the expressions similar to (62), (63), we obtain Z(X^{r−1}) = e ∈ D̂^{r}(n, m). □

5.3. Structure equations of B̂^{r}(M_{n} × U). Let T^{r′} T̂_{pr}^{ℝ(n+m,r)}(M′_{k} × U), r′ < r, be the pullback of the tangent bundle T T̂_{pr}^{ℝ(n+m,r′)}(M′_{k} × U) under the projection π_{r′}^{r} : T̂_{pr}^{ℝ(n+m,r)}(M′_{k} × U) → T̂_{pr}^{ℝ(n+m,r′)}(M′_{k} × U). We define fundamental semivector fields corresponding to elements V_{e}^{(r′)} ∈ T_{e}^{r′} B̂^{r}(ℝ^{n} × ℝ^{m}) on the bundles T̂_{pr}^{ℝ(n+m,r)}(M′_{k} × U) as sections of the bundles T^{r′} T̂_{pr}^{ℝ(n+m,r)}(M′_{k} × U) in the following way. Let Y_{0} = j^{r}f be an element of T̂_{pr}^{ℝ(n+m,r)}(M′_{k} × U) defined by a germ of morphism f : (ℝ^{n} × ℝ^{m}, (0, 0)) → (M′_{k} × U, (y_{0}, t_{0})), y^{i′} = f^{i′}(x^{j}, t^{a}), t′^{a} = t^{a} + t_{0}^{a}. Applying the functor T̂_{pr}^{ℝ(n+m,r)} to f, we obtain the germ

F : (B̂^{r}(M_{n} × U), e) → (T̂_{pr}^{ℝ(n+m,r)}(M′_{k} × U), Y_{0})

given by equations (83)

Y^{i′} = f^{i′}(x, t) + ∑_{p+s=1}^{r} \frac{1}{p!s!} \frac{∂^{|p+s|}f^{i′}}{∂x^{p}∂t^{s}} X^{∘}^{p}ν^{s}.  (97)

The jet j^{r}f defines the mapping

T^{r−1}T̂_{pr}^{ℝ(n+m,r)}(f) : T_{e}^{r−1}B̂^{r}(ℝ^{n} × ℝ^{m}) → T_{Y_{0}}^{r−1}T̂_{pr}^{ℝ(n+m,r)}(M′_{k} × U),

which assigns to V_{e}^{(r−1)} the value V^{(r−1)}(Y_{0}) of the corresponding fundamental semivector field V^{(r−1)} at Y_{0}. In terms of local coordinates, the field V^{(r−1)}(Y) is given by the equations (cf (51)

V^{i′}(Y) = ∑_{|p|+|s|=0}^{r−1}(p_{j} + 1)y_{p+j s}^{i′}ε^{p}ν^{s})V_{e}^{j} + ∑_{|p|+|s|=0}^{r−1}(s_{a} + 1)y_{p s+a}^{i′}ε^{p}ν^{s})v_{e}^{a}, (98)

v^{a}(Y) = v_{e}^{a},

where V_{e}^{j} ∈ ℝ(n + m, r), j = 1, …, n, and v_{e}^{a} ∈ ℝ, a = 1, …, m, are the standard coordinates of V_{e}^{(r−1)}. In particular, for M′_{k} = ℝ, the fundamental semivector field V^{(r−1)} corresponding to V_{e}^{(r−1)} can be regarded as the mapping

ℝ(n + m, r) ⊕ ℝ^{m} → ℝ(n + m, r − 1) ⊕ ℝ^{m},  (99)
(α = α_{ps}ε^{p}ν^{s}, t^{a}) ↦ (V_{e}^{j}∂_{j}(α) + v_{e}^{a}∂_{a}(α), v_{e}^{a}),

where ∂_{j} : ℝ(n + m, r) → ℝ(n + m, r − 1) and ∂_{a} : ℝ(n + m, r) → ℝ(n + m, r − 1) are the derivations defined, respectively, by ∂_{j}(ε^{i}) = δ_{j}^{i}, ∂_{j}(ν^{b}) = 0 and ∂_{a}(ε^{i}) = 0, ∂_{a}(ν^{b}) = δ_{b}^{a}. Using the derivations ∂_{j} and ∂_{a}, one can rewrite equations (98) in the form

V^{i′}(Y) = V_{e}^{j}∂_{j}(Y^{i′}) + v_{e}^{a}∂_{a}(Y^{i′}), v^{a}(Y) = v_{e}^{a}.  (100)

Note that fundamental semivector fields (99) on ℝ(n + m, r) ⊕ ℝ^{m} and (100) on T̂_{pr}^{ℝ(n+m,r)}(ℝ^{k} × ℝ^{m}) can be regarded as linear mappings, and so the bracket of fundamental semivector fields [V^{(r−1)}, W^{(r−1)}] induced by the Lie bracket of vector fields can be calculated as the bracket of linear endomorphisms. Taking into account that V^{i′}(Y) in (100) do not depend on t^{a}, and v^{a}(Y) are constant, from (100), we obtain [V^{(r−1)}, W^{(r−1)}] = U^{(r−2)}, where

U^{i′}(Y) = ((W_{e}^{k}∂_{k}V_{e}^{j} − V_{e}^{k}∂_{k}W_{e}^{j}) + (w_{e}^{b}∂_{b}V_{e}^{j} − v_{e}^{b}∂_{b}W_{e}^{j}))∂_{j}(Y^{i′}), (101)
u^{a}(Y) = 0.

We define the T_{e}^{r−2}B̂^{r}(ℝ^{n} × ℝ^{m})-valued 1-form θ̃^{r} on B̂^{r}(M_{n} × U) as follows: θ̃^{r} = π_{r−2}^{r−1} ∘ θ^{r}, where π_{r−2}^{r−1} : T_{e}^{r−1}B̂^{r}(ℝ^{n} × ℝ^{m}) → T_{e}^{r−2}B̂^{r}(ℝ^{n} × ℝ^{m}) is the canonical projection. Let θ̃^{r} = θ̃^{j}e_{j}^{(r−2)} + θ̃^{a}e_{a}^{(r−2)} be the expansion of θ̃^{r} in terms of the standard bases of the direct summands ℝ(n + m, r − 2)^{n} and ℝ^{m} in T_{e}^{r−2}B̂^{r}(ℝ^{n} × ℝ^{m}) ≡ ℝ(n + m, r − 2)^{n} ⊕ ℝ^{m}. Then, by calculations similar to those in Proposition 10, we obtain the following proposition.

Proposition 12. On the bundle B̂^{r}(M_{n} × U), the following structure equations hold:

dθ̃^{i} = θ^{j} ∧ ∂_{j} ∘ θ^{i} + θ^{a} ∧ ∂_{a} ∘ θ^{i}, dθ̃^{a} = 0.  (102)

Note 6. From the above discussion it follows that one can also take the space of (r − 1)-jets at zero of vector fields on ℝ^{n} × ℝ^{m} which project into constant vector fields on ℝ^{m} as the domain of values of the form θ̂^{r}.

Note 7. As in the case of the bundle B^{∞}(𝔸)T^{𝔸}M_{n} (see Note 5), passing to the projective limit, one can define the ℝ(n + m, ∞)^{n} ⊕ ℝ^{m}-valued structure form θ̂^{∞} on B̂^{∞}(M_{n} ×U). For r tending to infinity, the series of equations (102) gives the following expression for the exterior differential of the form θ̂^{∞}:

d(θ̂^{∞})^{i} = (θ̂^{∞})^{j} ∧ ∂_{j} ∘ (θ̂^{∞})^{i} + (θ̂^{∞})^{a} ∧ ∂_{a} ∘ (θ̂^{∞})^{i}, d(θ̂^{∞})^{a} = 0, (103)

where ∂_{j} and ∂_{a} are the derivations of the algebra ℝ(n + m, ∞) of formal power series in n + m variables ε^{i} and ν^{a} defined by the above indicated relations ∂_{j}(ε^{i}) = δ_{j}^{i}, ∂_{j}(ν^{b}) = 0 and ∂_{a}(ε^{i}) = 0, ∂_{a}(ν^{b}) = δ_{b}^{a}.

5.4. Connections in B̂^{r}(M_{n}×U). A connection Γ in a principal bundle P = P (M, G, π) is a G-invariant horizontal distribution on P [11], [14]. Since the horizontal planes at X ∈ P are in a bijective correspondence with the 1-jets of germs of sections s : (M, x) → (P, X), x = π(X), called also connection elements, a connection Γ can also be defined as a G-equi-variant section Γ : P → J^{1}P, where J^{1}P is the first jet prolongation of P (the set of all connection elements) [14].

Connection elements on B̂^{r}(M_{n} × U) are 1-jets of germs of sections γ : (M × U, (x_{0}, t_{0})) → B̂^{r}(M × U). In terms of the local coordinates (90), such a germ of section γ is given by equations

X^{∘}^{i} = ∑_{|p|+|s|=1}^{r} X_{ps}^{i}(x^{i}, t^{a})ε^{p}ν^{s},

and the corresponding connection element has coordinates

\begin{aligned}Γ_{j}^{i} & = \frac{∂ X^{∘}^{i}}{∂x^{j}} |_{(x_{0},t_{0})} = ∑_{|p|+|s|=1}^{r} \frac{∂ X^{∘}_{ps}^{i}}{∂x^{j}} |_{(x_{0},t_{0})}ε^{p}ν^{s} = ∑_{|p|+|s|=1}^{r} Γ_{psj}^{i}ε^{p}ν^{s} = ∑_{|p|=0}^{r} Γ_{pj}^{i}ε^{p}, \\ Γ_{a}^{i} & = \frac{∂ X^{∘}^{i}}{∂t^{a}} |_{(x_{0},t_{0})} = ∑_{|p|+|s|=1}^{r} \frac{∂ X^{∘}_{ps}^{i}}{∂t^{a}} |_{(x_{0},t_{0})}ε^{p}ν^{s} = ∑_{|p|+|s|=1}^{r} Γ_{psa}^{i}ε^{p}ν^{s} = ∑_{|p|=0}^{r} Γ_{pa}^{i}ε^{p} \end{aligned} ; (104)

where the coordinates Γ_{pj}^{i} and Γ_{pa}^{i} are elements of the quotient algebra ℝ(m, r)/𝔪(ℝ(m, r))^{|p|}. In what follows, following the standard summation convention, we will omit the sign of sum in sums like (104).

The right action Y = R_{Z}(X) of the group D̂^{r}(n, m) on B̂^{r}(M_{n} × U) induces the right action R_{Z}(j^{1}γ) = j^{1}(R_{Z} ∘ γ) of D̂^{r}(n, m) on the bundle of connection elements J^{1}B̂^{r}(M_{n} ×U). In terms of local coordinates, this action is of the form (see (91))

R_{Z} : (X^{i}, Γ_{pj}^{i}, Γ_{pa}^{i}) ↦ (Y^{i}, Γ_{pj}^{i}Z^{p}, Γ_{pa}^{i}Z^{p}).  (105)

Analytically, a connection Γ in B̂^{r}(M_{n} × U) can be given by the coordinates of the connection elements along the natural local section of B̂^{r}(M_{n} × U) defined by a local coordinate chart on M_{n} × U. These coordinates are functions Γ_{pj}^{i}(x^{i}, t^{a}), Γ_{pa}^{i}(x^{i}, t^{a}) of the local coordinates on M_{n}×U called the connection coefficients. The natural section is given by the equations X^{∘}_{pr}^{i} = δ_{p}^{i}δ_{r}^{0}, where δ_{p}^{i} = 1 when p_{i} = 1 and p_{j} = 0 if i ≠ j, and δ_{p}^{i} = 0 in all the other cases. Therefore, the connection elements at an arbitrary X ∈ B̂^{r}(M_{n} × U) are of the form

Γ_{j}^{i}(X) = Γ_{psj}^{i}(x^{j}, t^{a}) X^{∘}^{p}ν^{s}, Γ_{a}^{i}(Y) = Γ_{psa}^{i}(x^{j}, t^{a}) X^{∘}^{p}ν^{s}.

Coordinate transformations

x^{i′} = f^{i′}(x^{i}, t^{a}), t^{a′} = t^{a} + t_{0}^{a}

induce the following coordinate transformations on B̂^{r}(M_{n} × U):

X^{∘}^{i′} = ∑_{|p|=0}^{r} A_{p}^{i′}(x^{i}, t^{a}) X^{∘}^{p},  (106)

where A_{p}^{i′}(x^{i}, t^{a}) ∈ ℝ(m, r)/𝔪(ℝ(m, r))^{|p|} are defined by

A_{p}^{i′}(x^{i}, t^{a})ε^{p} = ∑_{|p|+|s|=1}^{r} \frac{1}{p!s!} \frac{D^{|p|+|s|}f^{i′}}{Dx^{p}Dt^{s}} ν^{s}ε^{p}.

Differentiating the coordinate transformations (106), one obtains the transformation law for the connection coefficients

Γ_{uj′}^{i′}(A_{s}^{∗}ε^{s})^{u} = \frac{∂x^{α}}{∂x^{j′}} (\frac{∂A_{p}^{i′}}{∂x^{α}} ε^{p} + A_{p}^{i′} ∑_{i=1}^{n} p_{i}Γ_{sj}^{i}ε^{p+s−i}) ,  (107)

\begin{aligned}Γ_{ua′}^{i′}(A_{s}^{∗}ε^{s})^{u} & = \frac{∂A_{p}^{i′}}{∂t^{a}} ε^{p} − \frac{∂x^{j}}{∂x^{j′}} \frac{∂x^{j′}}{∂t^{a}} \frac{∂A_{p}^{i′}}{∂x^{j}} ε^{p}+ \\ & + A_{p}^{i′} ∑_{i=1}^{n} p_{i} (Γ_{sa}^{i} − \frac{∂x^{j}}{∂x^{j′}} \frac{∂x^{j′}}{∂t^{a}} Γ_{sj}^{i}) ε^{p+s−i}, \end{aligned} (108)

where (A_{s}^{∗}ε^{s})^{u} = A^{u} for A^{i} = A_{s}^{i}ε^{s} and u = (u_{1}, …, u_{n}).

The horizontal distribution of a connection in B̂^{r}(M_{n} × U) is given by the equations

d Y^{∘}^{i} − Γ_{psj}^{i}(x^{j}, t^{a}) Y^{∘}^{p}ν^{s}dx^{j} − Γ_{psa}^{i}(x^{j}, t^{a}) Y^{∘}^{p}ν^{s}dt^{a} = 0 .  (109)

5.5. Associated connections in T̂_{σ}^{𝔸}(M_{n} × U). If one replaces a morphism (81) by a germ of isomorphism

\xymatrix{(ℝ^{n} × ℝ^{m}, (0, 0)) \ar[d] \ar[r]^{f} & (ℝ^{n} × ℝ^{m}, (0, 0)) \ar[d] \\ (ℝ^{m}, 0) \ar[r]^{id} & (ℝ^{m}, 0)}  (110)

then the corresponding diagram (82) gives the left action Φ_{σ} of the group

D̂^{q}(n, m) on the standard fiber T̂_{σ}^{𝔸}_{(0,0)}(ℝ^{n} × ℝ^{m}) ≡ T_{0}^{𝔸}ℝ^{n} ≡ 𝔸^{∘}^{n} of the

bundle T̂_{σ}^{𝔸}(M_{n} × U). If we consider T̂_{σ}^{𝔸}_{(0,0)}(ℝ^{n} × ℝ^{m}) as the set of 𝔸-ve-locities j^{𝔸}g of germs of the form

(see (84))

\xymatrix{(ℝ^{ℓ}, 0) \ar[r]^{g} \ar[d]_{id} & (ℝ^{n} × ℝ^{m}, (0, 0)) \ar[d] \\ (ℝ^{ℓ}, 0) \ar[r]^{σ̂} & (ℝ^{m}, 0)}

then the action

Φ_{σ} : D̂^{q}(n, m) × T̂_{σ}^{𝔸}_{(0,0)}(ℝ^{n} × ℝ^{m}) → T̂_{σ}^{𝔸}_{(0,0)}(ℝ^{n} × ℝ^{m})

is given by (85):

Φ_{σ} : (j^{q}f, j^{𝔸}g) ↦ j^{𝔸}(f ∘ g),  (111)

where (f, id) is a germ (110) of isomorphism in Mf^{m}. In the standard coordinates z_{ps}^{i} on D̂^{q}(n, m) and X^{∘}^{i} ∈ 𝔸^{∘} on T̂_{σ}^{𝔸}_{(0,0)}(ℝ^{n} × ℝ^{m}), the action (111), (Z, X) ↦ Y, is of the form

Y^{∘}^{i} = ∑_{|p|+|s|=1}^{q} z_{ps}^{i} X^{∘}^{p} σ^{∘}^{s}.  (112)

Note that, in general, the action Φ_{σ} is not effective. Relations (112) define an effective action on 𝔸^{∘}^{n} of some factor group D_{σ}(n, 𝔸) of the group D̂^{q}(n, m).

The action Φ_{σ} allows one to consider the bundle T̂_{σ}^{𝔸}(M_{n} × U) as associated with the frame bundle B̂^{q}(M_{n} × U). The mapping

Ψ_{σ} : B̂^{q}(M_{n} × U) × 𝔸^{∘}^{n} → T̂_{σ}^{𝔸}(M_{n} × U), (Y, Z^{∘}) ↦ X,

which defines on T̂_{σ}^{𝔸}(M_{n} × U) the structure of a bundle associated with B̂^{q}(M_{n} × U) has the following equations in terms of local coordinates:

X^{i} = ∑_{|p|+|s|=0}^{q} y_{ps}^{i} Z^{∘}^{p} σ^{∘}^{s}.  (113)

Fixing Z^{∘}^{i} in (113), from (109), we obtain the following equations of the horizontal distribution of the connection Γ on T̂_{σ}^{𝔸}(M_{n} × U):

d X^{∘}^{i} − Γ_{psj}^{i}(x^{j}, t^{a}) X^{∘}^{p} σ^{∘}^{s}dx^{j} − Γ_{psa}^{i}(x^{j}, t^{a}) X^{∘}^{p} σ^{∘}^{s}dt^{a} = 0 .

5.6. Examples. 1. The functor T̂_{0}^{𝔸} defined by the zero section σ = 0 : U → T^{𝔸}U, corresponding to the germ σ̂ = 0 (see (84)) at 0 ∈ ℝ^{ℓ} of the zero mapping 0 : ℝ^{ℓ} ∋ t ↦ 0 ∈ ℝ^{m}, is equivalent to the vertical Weil functor V^{𝔸}, the functor which assigns to p : M_{n} × U → U the bundle V^{𝔸}(M_{n} ×U) of 𝔸-velocities of germs g : (ℝ^{ℓ}, 0) → M_{n} ×U such that p∘g is a germ of constant mapping.

2. In the case when ℓ = m and σ̂ in (84) is the germ of the identity mapping id : ℝ^{m} → ℝ^{m} (and σ^{∘}^{a} = τ^{a}, a = 1, …, m, are the elements of the standard pseudobasis of 𝔸 ≡ T^{𝔸}ℝ), the functor T̂_{σ}^{𝔸} is equivalent to the functor T̂^{𝔸} studied in [2] which assigns to p : M_{n} ×U → U the bundle T̂^{𝔸}(M_{n} × U) of 𝔸-velocities of germs g ∘ tr_{t_{0}}, where g : (ℝ^{m}, t_{0}) → (M_{n} × U, (x, t_{0})) is a germ of section. In this case, the action (112) of the group D̂^{q}(n, m) on 𝔸^{∘}^{n} reduces to the action of the so-called 𝔸-affine differential group D_{n}(𝔸) [31], [34]:

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

The principal bundle associated with T̂^{𝔸}(M_{n} × U) is the bundle of 𝔸-affine frames B(𝔸)(M_{n} × U).

3. For the algebra ℝ(m, q), the group D_{n}(ℝ(m, q)) coincides with D̂^{q}(n, m) and the bundle B(ℝ(m, q))(M_{n} × U) coincides with B̂^{q}(M_{n} × U).

For m = 1 and q = 1, ℝ(1, 1) = ℝ(ε) is the algebra of dual numbers x + ẋε, ε^{2} = 0.

Local coordinates (x^{i} + ẋ^{i}ε, t) on T̂^{ℝ(ε)}(M_{n} × ℝ) are transformed as follows:

x^{i′} + ẋ^{i′}ε = φ^{i′}(x^{i}, t) + (\frac{∂φ^{i′}}{∂x^{i}} ẋ^{i} + \frac{∂φ^{i′}}{∂t} ) ε, t′ = t + t_{0}.

The standard fiber of T̂^{ℝ(ε)}(M_{n} × ℝ) is ℝ^{n}, and the structure group D_{n}(ℝ(ε)) is the group of affine transformations ẋ^{i′} = a_{i}^{i′}ẋ^{i} + a_{0}^{i′}. The horizontal distribution of a connection in B̂^{1}(M_{n} × ℝ) has equations [2]

dẏ^{i} = (Γ_{λα}^{i}(x, t)dx^{α} + Γ_{λ0}^{i}(x, t)dt)ẏ^{λ} + Γ_{0α}^{i}(x, t)dx^{α} + Γ_{00}^{i}(x, t)dt, (114)
dy_{μ}^{i} = (Γ_{λα}^{i}(x, t)dx^{α} + Γ_{λ0}^{i}(x, t)dt)y_{μ}^{λ} .  (115)

HIGHER ORDER GEOMETRY OF WEIL BUNDLES 103

Under a coordinate change on M_{n} × ℝ, the connection coefficients transform as follows:

Γ_{j′k′}^{i′} = \frac{∂x^{j}}{∂x^{j′}} \frac{∂x^{k}}{∂x^{k′}} (\frac{∂x^{i′}}{∂x^{i}} Γ_{jk}^{i} + \frac{∂^{2}x^{i′}}{∂x^{j}∂x^{k}} ) ,

Γ_{0k′}^{i′} = \frac{∂x^{k}}{∂x^{k′}} (\frac{∂x^{i′}}{∂x^{i}} Γ_{0k}^{i} + \frac{∂^{2}x^{i′}}{∂t∂x^{k}} − \frac{∂x^{j}}{∂x^{j′}} \frac{∂x^{j′}}{∂t} (\frac{∂x^{i′}}{∂x^{i}} Γ_{jk}^{i} + \frac{∂^{2}x^{i′}}{∂x^{j}∂x^{k}} )) ,

Γ_{k′0}^{i′} = \frac{∂x^{k}}{∂x^{k′}} (\frac{∂x^{i′}}{∂x^{i}} Γ_{k0}^{i} + \frac{∂^{2}x^{i′}}{∂t∂x^{k}} − \frac{∂x^{j}}{∂x^{j′}} \frac{∂x^{j′}}{∂t} (\frac{∂x^{i′}}{∂x^{i}} Γ_{kj}^{i} + \frac{∂^{2}x^{i′}}{∂x^{j}∂x^{k}} )) ,

Γ_{00}^{i′} = \frac{∂x^{i′}}{∂x^{i}} Γ_{00}^{i} + \frac{∂^{2}x^{i′}}{∂^{2}t} − \frac{∂x^{k}}{∂x^{k′}} \frac{∂x^{k′}}{∂t} (\frac{∂x^{i′}}{∂x^{i}} (Γ_{k0}^{i} + Γ_{0k}^{i}) + 2\frac{∂^{2}x^{i′}}{∂t∂x^{k}} +

+ \frac{∂x^{j}}{∂x^{j′}} \frac{∂x^{j′}}{∂t} (\frac{∂x^{i′}}{∂x^{i}} Γ_{kj}^{i} + 2\frac{∂^{2}x^{i′}}{∂x^{j}∂x^{k}} )) .

The equations of the horizontal distribution on T̂^{ℝ(ε)}(M_{n} × ℝ) corresponding to (114), (115) are of the form

dẋ^{i} = (Γ_{λα}^{i}(x, t)dx^{α} + Γ_{λ0}^{i}(x, t)dt)ẋ^{λ} + Γ_{0α}^{i}(x, t)dx^{α} + Γ_{00}^{i}(x, t)dt .

Along a section c : ℝ → M_{n} ×ℝ, the above Pfaff system gives the system of linear differential equations with respect to ẋ^{i}:

\frac{dẋ^{i}}{dt} = (Γ_{λα}^{i}(x, t)\frac{dx^{α}}{dt} + Γ_{λ0}^{i}(x, t))ẋ^{λ} + Γ_{0α}^{i}(x, t)\frac{dx^{α}}{dt} + Γ_{00}^{i}(x, t).

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Kazan State University, Kremlevskaya, 18, Kazan:420008, RUSSIA E-mail address: Galina.Bushueva@ksu.ru E-mail address: Vadim.Shurygin@ksu.ru

Received June 14, 2005