Connections of higher order and product preserving functors

Jacek Gancarzewicz (Cracow) Noureddine Rahmani (Oran) Modesto Salgado (Santiago de Compostela)

IMUJ PREPRINT 1997/21

Abstract

In this paper we consider a product preserving functor F of order r and a connection Γ of order r on a manifold M. We introduce horizontal lifts of tensor fields and linear connections from M to F(M) with respect to Γ. Our definitions and results generalize the particular cases of the tangent bundle and the tangent bundle of higher order.

0 Introduction

Let F be a product preserving functor (see [8] ), then FM is a fiber bundle, with standard fiber F_{0}(R^{n}), associated to the bundle L^{r}M of frames of order r, where n is the dimension of M and r is the order of F.

Tangent bundles, tangent bundles of higher order, tangent bundles of p^{r}-velocities, Weil bundles (bundles of infinitely near points) are examples of product preserving functors.

The horizontal lifts of tensor fields and linear connections to the tangents bundles, with respect to a linear connection, were introduced and studied in [19] and [20]. A similar study for the tangent bundles of higher order has been developped in [5] and [9] .

In this paper, we present the horizontal prolongations of tensor fields of type (1, 1) and linear connections from M to FM with respect to a connection Γ of order r on M, that is, a connection on the principal fibre bundle L^{r}M, which generalize the results given in [19] , [20] , [5] and [9] . Let us remark that we do not use local coordinates.

In Section 2 we recall some properties of such connections which are necessary for constructions given in the paper.

In Section 3 we recall λ-lifts of functions and a-lifts of vector fields. Next, we generalize these lifts for the sections of J^{r}(M, R) and J^{r−1}(T M).

In Sections 4 and 5 we the study relantionships between the horizontal fields and the a-lifts of vector fields. For that, we use the above lifts of sections and introduce a new family of vertical vector fields {f^{□}} on F(M), defined for each derivation f : J^{r}(M, R)_{0} ⟶ J^{r}(M, R)_{0}. The formulas, here obtained, are important since they allow us to study the properties of horizontal lifts of tensor fields and linear connections which are introduced in Sections 6 and 7 .

In Section 6 we introduce the horizontal lifts of tensor fields of type (1, 1) from M to F(M), with respect to Γ, and use these lifts to prolongate geometric structures from M to F(M).

In Section 7 we introduce the horizontal lift of a linear connection from M to F(M) with respect to Γ, and study the covariant derivation of horizontal lifts with respect to this new linear connection.

1 Product preserving functors

First we recall the definitions of product preserving functor.

A prolongation functor is a covariant functor F from the category of all manifolds and all mappings into the category of fibered manifolds satisfying the following conditions:

1. For every manifold M, F(M) is a fibered manifold over M and π = π_{M} : FM ⟶ M is its projection. For a point x ∈ M we denote by F_{x}(M) = π_{M}^{−1}(x) the fibre over x.

2. The naturality condition. For every mapping φ : M → M′ of two manifolds M, M′, for the induced mapping F(φ) : F(M) → F(M′) the following diagram

[figure: F(M); F(M′); F(φ); M; M; π_{M}; π_{M′}; φ]

commutes.

3. If φ : M → M′ is an embedding of two n-dimensional manifolds M, M′, then for x ∈ M the restriction F(φ)_{|F_{x}(M)} : F_{x}(M) → F_{φ(x)}(M′) is a diffeomorphism.

4. The regularity condition. If φ_{t} : M → M′ is a differentiable family of mappings, then F(φ_{t}) : F(M) → F(M′) is a differentiable family of mappings.

Kolář and Slovák have proved that the regularity condition is a consequence of conditions (1) — (3) of the definition (see [14]).

This definition immediately implies:

1. if U ⊂ M is an open subset then we can identify F(U) with F(M)_{|U} by F(i) : F(U) → F(M)_{|U}, where i : U → M is the inclusion;

2. F(R^{n}) is isomorphic with the trivial bundle R^{n} × F, where F = F_{0}(R^{n}). The isomorphism Ψ : R^{n} ×F → F(R^{n}) is given by Ψ(x, y) = F(τ_{x})(y), where τ_{x} : R^{n} → R^{n} is the translation.

For two manifolds M_{1}, M_{2} we denote by π_{i} : M_{1}×M_{2} → M_{i} the standard projection on the i-th factor, where i = 1, 2. A prolongation functor F is called a product preserving functor if for all manifolds M_{1}, M_{2} the mapping

(F(π_{1}), F(π_{2})) F(M_{1} × M_{2}) ⟶ F(M_{1}) × F(M_{2})

is a diffeomorphism.

For a product preserving functor F we will always identify F(M_{1} × M_{2}) with F(M_{1}) × F(M_{2}) by the diffeomorphism from the definition. After this identification we have

F(f_{1} × f_{2}) = F(f_{1}) × F(f_{2})
F(f, g) = (F(f), F(g))

for all mappings f_{1} : M_{1} → M′_{1}, f_{2} : M_{2} → M′_{2}, f : M → M′_{1} and g : M → M′_{2} .

The tangent bundle T M, the tangent bundle T^{r}M and the tangent bundle of p^{r}-velocities T_{p}^{r}M =J_{0}^{r}(R^{p}, M) (see [15, 16]) are important examples of product preserving functors. The more genaral functors are so-called Weil functors (see [13], [6], [7]).

Product preserving functors have many interesting properties. For all properties of product preserving functors we propose to consulte [13], [6], [7])

Now we recall so-called Weil algebra of F.

Proposition 1.1 If F is a product preserving functor then A = F(R) is a real associative, commutative and finite dimensional algebra.

If +, · : R^{2} → R are the addition and the multiplication on R and m_{α} : R → R is the multiplication by α ∈ R, then F(+), F(·), F(m_{α}) are the operations in A, F(0) and F(1) are the zero and the unity in A (after identification of constant mappings with their values).

The set N = F_{0}(R) is the ideal of nilpotent elements of A. We have A = R · 1 ⊕ N.

The algebra A = F(R) will be called Weil algebra of F. For a demostration see for instance [8], [7], [6], [13].

Let F be a product preserving functor and A = F(R) be its Weil algebra. Since the restriction of F to the category of one dimensional manifolds and thier embeddigs is a natural bundle, thus, by Palais-Terng theorem [17], it is of finite order, say r. This number r is called a order of F. Now it is easy to that for arbitrary manifolds M, M′ and arbitrary smooth mappings φ_{1}, φ_{2} : M → M′ and any point x ∈ M the following implication

j_{x}^{r}φ_{1} = j_{x}^{r}φ_{2} ⟹ F(j_{x}^{r}φ_{1}) = F(j_{x}^{r}φ_{2}) on F_{x}(M)

holds.

Two product preserving functors commute. We shall use this important property along this paper in the case of one of these two product preserving functors is the tangent bundle functor. Namely, for any product preserving functor F there exits a natural isomorphism η_{M} : F(T M) ⟶ T (FM) of vector bundles, over FM, which permets us to identify bundles F(T M) and T (FM) (for the details see [13], [6], [7]).

2 Connections of order r

The frame bundle of order r of M is the manifold L^{r}M whose points, called references of order r, are r-jets j_{0}^{r}γ at 0 ∈ R^{n} of diffeomorphisms γ of open neigborhoods of 0 ∈ R^{n} onto open subsets of M. Let π_{r} : L^{r}M → M be the target projection π_{r}(j_{0}^{r}γ) = γ(0). Then π_{r} : L^{r}M → M is a principal fibre bundle with structure group the manifold L_{n}^{r} of all r-jets j_{0}^{r}ξ of local diffeomorphisms ξ of R^{n} with the source and the target at 0 ∈ R^{n}. The right action on L^{r}M is given by

j_{0}^{r}γ · j_{0}^{r}ξ = j_{0}^{r}(γ ∘ ξ) .

If r = 1, L^{r}M = LM is the linear frame bundle of M.

A connection Γ of order r on M is a connection on the principal fiber bundle L^{r}M(M, L_{n}^{r}), Γ defines the right invariant horizontal distribution H^{r}, such that T (L^{r}M) = H^{r} ⊕ V(L^{r}M) where V(L^{r}M) denotes the vertical distribution.

Let E be a fiber bundle associated with L^{r}M with standard fibre V, corresponding to a left action λ of L_{n}^{r} on V. A connection of order r on M defines a decomposition of the tangent bundle T E

T E = H ⊕ V(E)

where V(E) is the vertical distribution and H is the so-called also horizontal distribution, given by H_{Ψ(p,z)} = d_{p}Ψ_{z}(H_{p}^{r}) , being Ψ : L^{r}M × V ⟶ E the canonical map for this associated bundle and Ψ_{z} : L^{r}M ⟶ E the mapping defined by Ψ_{z}(q) = Ψ(q, z), q ∈ L^{r}M, z ∈ V.

If X is a vector field on M, the horizontal lift of X to E with respect to a connection of order r is the vector field X^{H} on E defined by

X^{H}(y) = (d_{y}π|H_{y})^{−1}(X_{π(y)}) y ∈ E, (2.1)

where π : E → M is the projection.

A connection of order r defines a covariant derivation of sections of any vector bundle associated with L^{r}M. We recall this definition.

Let E be a vector bundle associated to L^{r}M with standard fibre V and left action λ of L_{n}^{r} on V (in this case V is a vector space and each left translation λ_{a} on V is linear). If σ : M ⟶ E is a section of E, we can consider a mapping σ̃ : L^{r}M ⟶ V given by

σ̃(p) = (p̃^{−1} ∘ σ ∘ π_{r})(p) , (2.2) where π_{r} : L^{r}M ⟶ M is the bundle projection and p̃ : V ⟶ E_{π(p)} is the linear isomorphism between the standard fibre V and the fiber E_{π(p)} of E over π_{r}(p) given by p̃(z) = Ψ(p, z). This mapping σ̃ will be called Crittenden’s mapping associated with σ.

The mapping σ̃ satisfies the condition σ̃ ∘R_{a} = λ_{a^{−1}} ∘σ̃, for each element a ∈ L_{n}^{r}, where R_{a} is the right translation on L^{r}M. We know (see Crittenden [2]) that for a mapping A : L^{r}M ⟶ V satisfying the condition A ∘ R_{a} = λ_{a^{−1}} ∘ A for every element a ∈ L_{n}^{r} there exits a unique section σ : M ⟶ E such that A = σ̃ .

If X is a vector field on M and σ : M ⟶ E is a section, then the covariant derivative D_{X}σ is the unique section D_{X}σ : M ⟶ E corresponding to the mapping X^{H^{r}}(σ̃) : L^{r}M ⟶ V (see [2]), where X^{H^{r}} denotes the horizontal lift of X to L^{r}M. According to formula (2.2) we have

D_{X}σ(π(p)) = (p̃ ∘ X^{H^{r}}(σ̃)(p) = Ψ(p, X^{H^{r}}(σ̃))(p)) . (2.3)

It is easy to verify that the covariant derivation has the following properties

\{ \begin{matrix}D_{X+X′}σ & = & D_{X}σ + D_{X′}σ , \\ D_{X}(σ + σ′) & = & D_{X}σ + D_{X}σ′ , \\ D_{fX}σ & = & fD_{X}σ , \\ D_{X}(fσ) & = & X(f)σ + fD_{X}σ . \end{matrix} (2.4)

for all vector fields X, X′, all sections σ, σ′ and each function f on M.

Now we can define the curvature of a connection of order r. We consider the mapping

\begin{matrix}R : \underline{T M} × \underline{T M} × \underline{E} ⟶ \underline{E} , \\ R(X, Y)σ = (D_{X} ∘ D_{Y} − D_{Y} ∘ D_{X} − D_{[X,Y]})(σ) , \end{matrix}

where denotes the C^{∞}(M)-module of sections of the corresponding bundle.

From (2.4) we deduce that R is C^{∞}(M)-linear. Then R can be considered as a mapping

R : T M ×_{M} T M ×_{M} E ⟶ E

such that its restriction to each fiber is 3-linear over R or, equivalently, R is an element of T^{∗}M ⊗ T^{∗}M ⊗ E^{∗} ⊗ E.

R is called the curvature of the connection, and the mapping R(X, Y) : E ⟶ E given by σ ∈ E ⟶ R(X, Y)σ ∈ E is called the curvature transformation corresponding to the vector fields X and Y on M.

Let us observe that in the case r = 1 and E = T M, we obtain the definition of curvature and curvature transformation of a linear connection.

In this paper we will use the covariant derivation of sections of three vector bundles of order r associated to L^{r}M. Namely we will use it in the following cases: (1) the (r − 1)-jet prolongation of the tangent bundle J^{r−1}T M , (2) the r-order cotangent bundle T^{r∗}M, and (3) the extended rorder cotangent bundle J^{r}(M, R). We recall the definitions of these vector bundles

J^{r−1}T M = { j_{x}^{r−1}X | X is a vector field, x ∈ M } ,
T^{r∗}M = { j_{x}^{r}f | f is a function on M: f(x) = 0, x ∈ M } ,
J^{r}(M, R) = { j_{x}^{r}f | f is a function on M }
= T^{r∗}M × R .

If r = 1 then J^{r−1}T M = T M is the tangent bundle and T^{r∗}M = T^{∗}M is the cotangent bundle.

The standard fibres of these vector bundles are J_{0}^{r−1}T R^{n}, J_{0}^{r}(R^{n}, R)_{0} and J_{0}^{r}(R^{n}, R), respectively, and the left actions of the structure group L_{n}^{r} on the corresponding standard fibers are given by

j_{0}^{r}ξ · j_{0}^{r−1}X = j_{0}^{r−1}(d_{0}ξ ∘ X)

in the first case, and

j_{0}^{r}ξ · j_{0}^{r}f = j^{r}(f ∘ ξ^{−1})

in the second and third cases.

Of course, the covariant derivations of sections of these vector bundles have properties (2.4), but in the case of the r-order cotangent bundle we can find an additional property of the covariant derivation.

The r-order cotangent bundle T^{r∗}M is a vector bundle but in this case we can define an algebra structure on each fiber of T^{r∗}M. Namely, if j_{x}^{r}f and j_{x}^{r}g are two element of the same fibre T_{x}^{r∗}M then we can define its product setting

(j_{x}^{r}f) (j_{x}^{r}g) = j_{x}^{r}(fg) .

These algebra stuctures on fibres define a structure of algebra on sections \underline{T^{r∗}M}. We know (see [3]) that for each vector field X the mapping D_{X} is an algebra derivation, e.g. for all sections σ, σ′ ∈ \underline{T^{r∗}M} we have

D_{X}(σ σ′) = D_{X}(σ) σ′ + σ D_{X}(σ′) . (2.5)

A similar property can be formulated for the extended r-order tangent bundle.

The covariant derivation of sections of the r-order cotangent bundle T^{r∗}M is important in the theory of connections of order r because for an operator D satisfying (2.4) and (2.5) there exists a unique connection of order r on M such that D is its covariant derivation (see [3]).

Let us observe that for r = 1 the algebra structure on T^{r∗}M = T^{∗}M is trivial, that is vw = 0 for all v, w ∈ T_{x}^{∗}M, but formula (2.5) is not trivial for each r ≥ 2.

As an immediate consequence of the definition of R(X, Y) we have the following

Proposition 2.1 For each vector fields X, Y on M the curvature transformation R(X, Y) : T^{r∗}M ⟶ T^{r∗}M is an algebra derivation.

From (2.3) we easily deduce

Proposition 2.2 Let E be a vector bundle associated with L^{r}M and a standard fibre V. Let X, Y be vector fields on M and σ be a section of E. If σ̃ and R̃(X, Y)σ denote, respectively, the associated mappings from L^{r}M to V with σ and R(X, Y)σ, then we have

R̃(X, Y)σ = ([X^{H^{r}}, Y^{H^{r}}] − [X, Y]^{H^{r}})σ̃.

3 Lifts of functions and vector fields to product preserving functors

In this section we recall some fundamental results of product preserving functorthat we shall use in the next sections, for details see [6], [8], [7].

Let F(M) be a product preserving functor and A = F(R) be its Weil algebra. If f : M → R is a function and λ : A → R is a smooth function, then the λ-lift of f to F(M) is the function on F(M) defined by

f^{(λ)} = λ ∘ F(f) . (3.1)

It is easy to verify that f^{(λ)} is a smooth function on F(M)

This family of functions on F(M) characterizes vector fields on F(M). Namely we have:

Proposition 3.1 Let F be a product preserving functor and A = F(R) be its Weil algebra.

If X̃, Ỹ are vector fields on F(M) such that for every smooth function f on M and every linear function λ : A → R we have X̃(f^{(λ)}) = Ỹ (f^{(λ)}) , then X̃ = Ỹ.

We can define an A-module structure of fibers of T (FM). Let m : R × T M ⟶ T M be the multiplication by scalars, e.g., m(t, v) = tv. The induced mapping F(m) : A × F(T M) ⟶ F(T M) defines an A-module structure on fibres of T (FM) by the formula

a · ṽ = η_{M}(F(m)(a, η_{M}^{−1}(ṽ))

for a ∈ A and ṽ ∈ T (FM), where η_{M} : F(T M) ⟶ T (FM) is the natural isomorphism which commutes a product preserving functor F with the tangent bundle functor [12].

If X : M → T M is a vector field on M, for each a ∈ A, the a-lift X^{(a)} of X to F(M) is the vector vector field on F(M) defined by the formula

X^{(a)} = a · η_{M}(F(X)) .

We know that X^{(a)} is the unique vector field on F(M) such that

X^{(a)}(f^{(λ)}) = (Xf)^{(λ∘l_{a})} (3.2)

where l_{a} : A → A is the translation l_{a}(b) = ab (see [6], [8]). We must remark that X^{(1)} is the complete lift X^{C} (see also [12]). From (3.2) we can deduce properties of a-lifts. Namely we have

Proposition 3.2 Let F be a product preserving functor and A = F(R) be its Weil algebra. If X, Y are vector fields on M, α, β ∈ R and a, b ∈ A, then

(α X + β Y)^{(a)} = α X^{(a)} + β Y^{(a)}
X^{(αa+βb)} = αX^{(a)} + βX^{(b)}
[X^{(a)}, Y^{(b)}] = [X, Y]^{(ab)}.

From results of Kolář [12] we can deduce that for a ∈ N the a-lift of a vector field X to a product preserving functor F of order r depends only on (r − 1)-jet of X. More precisely, if X and Y are two vector fields on M, then the following implication

j_{x}^{r−1}X = j_{x}^{r−1}Y ⟹ X_{|F_{x}M}^{(a)} = Y_{|F_{x}M}^{(a)} (3.3) holds. This remark permits to define a-lifts to F(M) of sections to the

J^{r−1}(T M) = { j_{x}^{r−1}X : x ∈ M, X is a vector field on M } ,

when a is a nilpotent element.

If a ∈ N and σ : M → J^{r−1}(T M) is a smooth section of J^{r−1}(T M), we define a vector field σ^{(a)} on F(M) setting for each point y ∈ F(M)

σ^{(a)}(y) = X^{(a)}(y) , (3.4)

where X is a vector field on M such that σ(π(y)) = j_{π(y)}^{r−1}X . From (3.3) we deduce that σ^{(a)} is well defined. Of course, if σ : M → J^{r−1}(T M) is a smooth section and a ∈ N, then σ^{(a)} is a smooth vector field on F(M).

From Proposition 3.2 and definition (3.4) we deduce immediately:

Proposition 3.3 Let F be a product preserving functor and A = F(R) be its Weil algebra. If σ_{1}, σ_{2} are sections of J^{r−1}(T M), α, β ∈ R and a, b ∈ N, then

(α σ_{1} + β σ_{2})^{(a) ,} = α σ_{1}^{(a)} + β σ_{2}^{(a)}
σ_{1}^{(αa+βb)} = ασ_{1}^{(a)} + βσ_{1}^{(b) .}

If X is a vector field on M, then

(j^{r−1}X)^{(a)} = X^{(a)} , (3.5)

where j^{r−1}X : M ⟶ J^{r−1}(T M) is the section defined by (j^{r−1}X)(x) = j_{x}^{r−1}X for x ∈ M.

The a-lifts of vector fields have the following interesting property.

Proposition 3.4 Let F be a product preserving functor and A = F(R) be its Weil algebra. Then the family { X^{(a)} }, where X is a vector field on M and a ∈ A, generates the module of vector fields on F(M). The family { X^{(a)} }, where X is a vector field on M and a is a nilpotent element in A, generates the module of vertical vector fields on F(M).

Finally, if ∇ is a linear connection on M, we can consider a linear connection ∇^{C} on F(M) (see [8], [6]). ∇^{C} is the unique linear connection on F(M) such that for all vector fields X, Y on M and all a, b ∈ A we have

∇_{X^{(a)}}^{C}Y^{(b)} = (∇_{X}Y)^{(ab)} . (3.6)

4 Horizontal lift of vector fields to a product preserving functor

Let F be a product preserving functor of order r and A = F(R) be its Weil algebra . Then F(M) is an associated bundle with the r-frame bundle L^{r}M and with standard fibre F_{0}(R^{n}), where n = dim M. Let Ψ : L^{r}M × F_{0}(R^{n}) ⟶ F(M) be the canonical mapping and let Γ be a connection of order r on M.

The horizontal lift of vector fields defined in Section 2 is natural in the following sense:

Proposition 4.1 Let con(M, r) be the set of all r-order connections on M and let X (M) be the space of vector fields on M. If L_{M} denotes the mapping

L_{M} : (Γ, X) ∈ con(M, r) × X (M) ⟶ X^{H} ∈ X (FM)

then for every mapping φ : M → M′ and for every X ∈ X (M), X′ ∈ X (M′), Γ ∈ con(M, r), Γ′ ∈ con(M′, r) the following implication

Xt and X′ are φ − related , Γ and Γ′ are φ − related ⇓

L_{M}(X, Γ) and L_{M′}(X′, Γ′) are F(φ) − related

holds.

From general theory of principal fiber bundles, it is well-known that the 1-parameter group of transformations of the horizontal lift of a vector field is characterized by the following proposition.

Proposition 4.2 Let F be a product preserving functor of order r and Γ be an r-order connection on M.

Let X be a smooth vector field on M and let φ_{t} be its local 1-parameter group of transformations defined on an open subset U ⊂ M. If for a point y ∈ F(M)_{|U} we denote by t ⟶ φ̂_{t}(y) the unique horizontal curve such that π ∘ φ̂_{t} = φ_{t} ∘ π and φ̂_{0}(y) = y, then the family φ̂_{t} is a local 1-parameter group of transformations of X^{H}.

If φ̃_{t} is a local 1-parameter group of transformations of X^{H^{r}} then we have φ̂_{t}(Ψ(p, z)) = Ψ(φ̃_{t}(p), z) , (4.1)

where X^{H^{r}} is the horizontal lift of X to the r-frame bundle L^{r}M.

In order to charaterize X^{H}, according to Proposition 3.1 we must compute X^{H}(f^{(λ)}).

First of all, we define the λ-lift of sections of J^{r}(M, R). Let σ : M ⟶ J^{r}(M, R) be a section. If λ : A ⟶ R is a linear function then we define the λ-lift σ^{(λ)} of σ , as the function on F(M) defined by

σ^{(λ)}(y) = f^{(λ)}(y) = (λ ∘ F(f))(y) , (4.2)

where f is a function on M such that σ(π(y)) = j_{π(y)}^{r}f. Since F is of order r, F(f)(y) depends only on j_{π(y)}^{r}f and therefore σ^{(λ)} is well defined.

Proposition 4.3 If X is a vector field on M, f is a function on M and λ : A ⟶ R is a linear function, then

X^{H}(f^{(λ)}) = (D_{X}j^{r}f)^{(λ)} ,

where j^{r}f : x ∈ M ⟶ j_{x}^{r}f ∈ J^{r}(M, R) is the section defined by f.

PROOF. The canonical mappings for the associated bundles F(M) and J^{r}(M, R) are given, respectively, by

Ψ : L^{r}M × F_{0}(R^{n}) ⟶ F(M)

Ψ(j_{0}^{r}γ, z) = F(γ)(z) , (4.3)

Φ : L^{r}M × J_{0}^{r}(R^{n}, R) ⟶ J^{r}(M, R)

Φ(j_{0}^{r}γ, j_{0}^{r}f) = j_{γ(0)}^{r}(f ∘ γ^{−1}) . (4.4)

Since we shall prove the formula locally, without lost of generality we can assume that L^{r}M is a trivial bundle.

Let σ : M ⟶ L^{r}M be a section. Let φ_{t} be an 1-parameter group of transformations of X and let φ̂_{t} and φ̃_{t} be the 1-parameter groups of transformations of X^{H} and X^{H^{r}}, respectively. According to (4.1) we have

(φ̂_{t} ∘ Ψ)(σ(x), z) = Ψ(φ̃_{t}(σ(x)), z) , (4.5)

where x ∈ M and z ∈ F_{0}(R^{n}). For every point x ∈ M and every t the r-frames φ̃_{t}(σ(x)) and σ(φ_{t}(x)) belong to the same fiber, thus there exists an element a_{t,x} ∈ L_{n}^{r} such that

φ̃_{t}(σ(x)) = σ(φ_{t}(x)) · a_{t,x} . (4.6)

Let us suppose that

σ(x) = j_{0}^{r}γ_{x} , a_{t,x} = j_{0}^{r}ξ_{t,x} , (4.7)

where γ_{x} : R^{n} ⟶ M , γ_{x}(0) = x and ξ_{t,x} : R^{n} ⟶ R^{n}, ξ_{t,x}(0) = 0. Using this notation, from (4.5), (4.6) and (4.3) we obtain

φ̂_{t}(Ψ(σ(x), z) = Ψ(j_{0}^{r}(γ_{φ_{t}(x)} ∘ ξ_{t,x}), z) = F(γ_{φ_{t}(x)} ∘ ξ_{t,x})(z) . (4.8)

If y ∈ FM and x = π(y) ∈ M, then there exists an element z ∈ F_{0}(R^{n}) such that y = Ψ(σ(x), z). Now, using (4.8) we have

X^{H}(f^{(λ)})(y) = \frac{d}{dt}(f^{(λ)}(φ̂_{t}(y)))_{|t=0} = \frac{d}{dt}(f^{(λ)}(φ̂_{t}(Ψ(σ(x), z))))_{|t=0}
= \frac{d}{dt}(λ ∘ F(f ∘ γ_{φ_{t}(x)} ∘ ξ_{t,x})(z))_{|t=0}
= λ ∘ F(\frac{d}{dt}(f ∘ γ_{φ_{t}(x)} ∘ ξ_{t,x})_{|t=0})(z) ,

because λ and the mapping f ∈ C^{∞}(M, R) → F(f) ∈ C^{∞}(F(M), A) are linear. Thus we have

X^{H}(f^{(λ)})(y) = λ ∘ F(\frac{d}{dt}(f ∘ γ_{φ_{t}(x)} ∘ ξ_{t,x})_{|t=0})(z) . (4.9)

On the other hand, the Crittenden’s mapping

j̃^{r}f : L^{r}M ⟶ J_{0}^{r}(R^{n}, R)

is given by j̃^{r}f(j_{0}^{r}γ) = j_{0}^{r}(f ∘ γ). Then

X^{H^{r}}(j̃^{r}f)(σ(x)) = \frac{d}{dt}(j̃^{r}f(φ̃_{t}(σ(x))))_{|t=0}
= \frac{d}{dt}(j_{0}^{r}(f ∘ γ_{φ_{t}(x)} ∘ ξ_{t,x}))_{|t=0}
= j_{0}^{r}(\frac{d}{dt}(f ∘ γ_{φ_{t}(x)} ∘ ξ_{t,x})_{|t=0}) ,

because φ̃_{t}(σ(x)) = j_{0}^{r}(γ_{φ_{t}(x)} ∘ ξ_{t,x}) and j^{r}( ) is linear. Thus, from (3.3) and (5.4) we obtain

(D_{X}j^{r}f)(x) = j_{0}^{r}(\frac{d}{dt}(f ∘ γ_{φ_{t}(x)} ∘ ξ_{t,x})_{|t=0} ∘ γ_{x}^{−1}) ,

and from above definition of λ-lift of sections we have

(D_{X}j^{r}f)^{(λ)}(y) = (λ ∘ F(\frac{d}{dt}(f ∘ γ_{φ_{t}(x)} ∘ ξ_{t,x})_{|t=0} ∘ γ_{x}^{−1}))(y)
(4.10)  = (λ ∘ F(\frac{d}{dt}(f ∘ γ_{φ_{t}(x)} ∘ ξ_{t,x})_{|t=0}))(z) ,

because y = Ψ(σ(x), z) = F(γ_{x})(z). From (4.9) and (4.10) we deduce our proposition.□

5 The bracket of vector fields on F(M).

We shall study prolongations of geometrical structures from a manifold M to a product preserving functor F and also the integrability of the prolonged structures. To do this we shall need formulas for the bracket of different vector fields on F(M). From Proposition 3.2 we know the bracket of a-lifts, but we shall need also to know formulas for [X^{H}, Y^{(a)}] and [X^{H}, Y^{H}].

In order to compute [X^{H}, Y^{H}], we shall associate to each derivation f of T^{r∗}M a vertical vector field f□ on F(M). We shall define f□ in a few steps.

Let us recall that T^{r∗}M is a vector bundle with standard fibre V = J_{0}^{r}(R^{n}, R)_{0} associated to L^{r}M and left action λ_{a} : V ⟶ V given by

λ_{a}(j_{0}^{r}Z) = j^{r}(Z ∘ ξ^{−1}) , a = j_{0}^{r}ξ ∈ L_{n}^{r} , j_{0}^{r}Z ∈ V .

Let p = j_{0}^{r}γ ∈ L^{r}M and let p̃ be the isomorphism between V and the fibre of T^{r∗}M over π(p), given by

p̃: j_{0}^{r}Z ∈ V ⟶ j_{γ(0)}^{r}(Z ∘ γ^{−1}) ∈ (T^{r∗}M)_{π(p)} . (5.1)

If f : T^{r∗}M ⟶ T^{r∗}M is a derivation of T^{r∗}M, for each point p ∈ L^{r}M we define a derivation of V by

f_{p} = p̃^{−1} ∘ f_{π(p)} ∘ p̃: V ⟶ V , (5.2)

where f_{π(p)} denotes the restriction of f to the fiber over π(p). From this definition we deduce

Lemma 5.1 For p ∈ L^{r}M and a ∈ L_{n}^{r} we have f_{p·a} = λ_{a^{−1}} ∘ f_{p} ∘ λ_{a} .

Let us observe that we can consider the structural group L_{n}^{r} as an open subset of V^{n} = V × ⋯ × V in the following natural way

\begin{matrix}L_{n}^{r} & ↪ & V^{n} \\ j_{0}^{r}ξ = j_{0}^{r}(ξ^{1}, …, ξ^{n}) & ↪ & (j_{0}^{r}ξ^{1}, …, j_{0}^{r}ξ^{n}) , \end{matrix}

so the left translation L_{a} and the right translation R_{a} of L_{n}^{r} on L_{n}^{r} can be prolongated to V^{n}. We shall denote these prolongations by the same symbols, so

L_{a}(j_{0}^{r}Z) = j_{0}^{r}(ξ ∘ Z) R_{a}(j_{0}^{r}Z) = j_{0}^{r}(Z ∘ ξ) ,

where j_{0}^{r}Z ∈ V^{n} and a = j_{0}^{r}ξ ∈ L_{n}^{r} .

Let us observe that R_{a} : V^{n} ⟶ V^{n} is linear for each r and L_{a} : V^{n} ⟶ V^{n} is not linear for r ≥ 2. From definitions of R_{a} and λ_{a} we deduce

λ_{a} × ⋯ × λ_{a} = R_{a^{−1}} . (5.3)

If f : T^{r∗}M ⟶ T^{r∗}M is a derivation and p ∈ L^{r}M we define

f̃_{p} = f_{p} × ⋯ × f_{p} : V^{n} ⟶ V^{n} . (5.4)

From (5.3) and (5.4) we deduce the following lemma.

Lemma 5.2 For p ∈ L^{r}M and a ∈ L_{n}^{r} we have f̃_{p·a} = R_{a} ∘ f̃_{p} ∘ R_{a^{−1}} .

Lemma 5.3 If f is a derivation of T^{r∗}M, p ∈ L^{r}M and a ∈ L_{n}^{r} ⊂ V^{n}, then

f̃_{p}(a) = d_{e}L_{a}(f̃_{p}(e)) ,

where e = j_{0}^{r}(id) is the unit element of L_{n}^{r}.

PROOF. In order to simplify the formulas we shall use along the proof the following notations: f = f_{p} , f̃ = f̃_{p} .

Since f : V ⟶ V is a derivation, for any element z ∈ V and any natural integer k we have

f(z^{k}) = kz^{k−1}f(z) .

For ζ = (z_{1}, …, z_{n}) ∈ V^{n} and a n-tuple α = (α_{1}, …, α_{n}), where α_{i} is a non-negative integers, if we denote by ζ^{α} = z_{1}^{α_{1}} … z_{n}^{α_{n}} , then from the last formula we deduce

f(ζ^{α}) = ∑_{i=1}^{n} α_{i} z_{1}^{α_{1}} … z_{i−1}^{α_{i−1}} z_{i}^{α_{i}−1}z_{i+1}^{α_{i+1}} … z_{n}^{α_{n}}f(z_{i})

(5.5)  = ∑_{i=1}^{n} α_{i} ζ^{α−e_{i}} f(z_{i}) .

where e_{i} = (0, …, 1, …, 0), with 1 at i-th position.

If q^{i} = j_{0}^{r}(u → u^{i}) be the r-jet of standard projection on the i-th factor, then the unit element e of L_{n}^{r} can be written in the form e = (q^{1}, …, q^{n}), and with the above notation we have

e^{α} = (q^{1})^{α_{1}} … (q^{n})^{α_{n}} .

Let j_{0}^{r}Z = j_{0}^{r}(Z^{1}, …, Z^{n}) ∈ V^{n}, where Z^{i} is a function on R^{n}. We can assume that Z^{i} is a polynomial of degree r, that is,

Z^{i}(u) = Z^{i}(u^{1}, …, u^{n}) = ∑_{1≤|α|≤r} A_{α}^{i}u^{α} , (5.6)

where u^{α} = (u^{1})^{α_{1}} … (u^{n})^{α_{n}} , and |α| = α_{1} + ⋯ + α_{n}.

Now, using this notation we have

j_{0}^{r}Z^{i} = ∑_{1≤|α|≤r} A_{α}^{i}e^{α}

and from (5.5) we obtain

f(j_{0}^{r}Z^{i}) = ∑_{1≤|α|≤r} ∑_{j=1}^{n} α_{j} A_{α}^{i} e^{α−e_{j}}f(q^{j}) . (5.7)

If we put f(q^{j}) = j_{0}^{r}Q^{j} then the last formula can be written as follows

f(j_{0}^{r}Z^{i}) = ∑_{1≤|α|≤r} ∑_{j=1}^{n} j_{0}^{r}(u → α_{j} A_{α}^{i} u^{α−e_{j}}Q^{j}(u)) (5.8)

and we have

f̃(e) = (f(q^{1}), …, f(q^{n})) = (j_{0}^{r}(Q^{1}), …, j_{0}^{r}(Q^{n}))
(5.9)  = j_{0}^{r}Q ,

where Q = (Q^{1}, …, Q^{n}).

Since V^{n} is a vector space we can identify V^{n} with its tangent space T_{e}(V^{n}) by the mapping

X ∈ V^{n} ⟶ \frac{d}{dt}(e + tX)_{|t=0} ∈ T_{e}V^{n} .

So we have

d_{e}L_{a}(f̃(e)) = d_{e}L_{a}(\frac{d}{dt}(e + tf̃(e))_{|t=0})
= \frac{d}{dt}(L_{a}(e + tf̃(e))_{|t=0}

If we denote

d_{e}L_{a}(f̃(e)) = ((d_{e}L_{a}(f̃(e)))^{(1)}, …, (d_{e}L_{a}(f̃(e)))^{(n)})

then if a = j_{0}^{r}ξ we have

(d_{e}L_{a}(f̃(e)))^{(i)} = \frac{d}{dt}(j_{0}^{r}(u → ξ^{i}(u + tQ(u)))_{|t=0}
= j_{0}^{r}(u → \frac{d}{dt}(ξ^{i}(u + tQ(u)))_{|t=0})
= j_{0}^{r}(u → \frac{d}{dt}( ∑_{1≤|α|≤r} A_{α}^{i}(u + tQ(u))^{α}))_{|t=0})

= j_{0}^{r}( ∑_{1≤|α|≤r} ∑_{j=1}^{n} u → A_{α}^{i} α_{j} u^{α−e_{j}}Q^{j}(u)) .

From the last formula and (5.8) we deduce our lemma because we have

d_{e}L_{a}(f̃(e) = ((d_{e}L_{a}(f̃(e)))^{(1)}, …, d_{e}L_{a}(f̃(e)))^{(n)})
f̃(ξ) = (f(j_{0}^{r}ξ^{1}), …, f(j_{0}^{r}ξ^{n})) .□

Let us denote by l_{n}^{r} the Lie algebra of L_{n}^{r}, that we can identify (as a vector space) with V^{n}. Now if f : T^{r∗}M ⟶ T^{r∗}M is a derivation, we define a function E (f) : L^{r}M ⟶ l_{n}^{r}, where l_{n}^{r} is the Lie algebra of the structure group L_{n}^{r} by

E (f)(p) ≡ E (f, p) = f̃_{p}(e) . (5.10)

Lemma 5.4 For an element a ∈ L_{n}^{r} we have

E (f) ∘ R_{a} = Ad_{a^{−1}} ∘ E (f) ,

where Ad_{a^{−1}} = d_{e}ad_{a^{−1}} : l_{n}^{r} ⟶ l_{n}^{r} is the adjoint endomorphism of the Lie algebra and R_{a} : L^{r}M → L^{r}M is the right traslation on L^{r}M.

PROOF. Ii is a direct consequence of Lemmas 5.2 and 5.3 taking into account that for the linear mapping R_{a} we have dR_{a} = R_{a}. □

Now we can define the vertical vector field f□ on F(M). Let y = Ψ(p, z) ∈ F(M) be a point of F(M), where Ψ : L^{r}M × F_{0}(R) ⟶ F(M) is the canonical mapping for this associated bundle. We define

f□(y) = d_{p}Ψ_{z}( E (f, p)_{p}^{∗}) , (5.11)

where E (f, p)^{∗} denotes the fundamental vertical vector field on L^{r}M defined by E (f, p). From Lemma 5.4 and the well known formula dR_{a}A^{∗} = (Ad(a^{−1})A)^{∗}, A ∈ l_{n}^{r}, we can easily deduce that f□ is well defined.

Now we can establish the formula for the bracket [X^{H}, Y^{H}].

Proposition 5.5 If X, Y are vector fields on M then

[X^{H}, Y^{H}] = [X, Y]^{H} + (R(X, Y))□ , (5.12)

where R(X, Y) : T^{r∗}M ⟶ T^{r∗}M is the curvature transformation in the r-order cotangent bundle.

PROOF. From Proposition 2.1 we have that R(X, Y) is a derivation of T^{r∗}M, so we can consider the vector field (R(X, Y))□.

We fix a point y = Ψ(p, z) of F(M). Since X^{H^{r}} is projectable vector field with projection X, then the vector field [X^{H^{r}}, Y^{H^{r}}] − [X, Y]^{H^{r}} is a vertical vector field on L^{r}M. Then there exists an element Z of l_{n}^{r} such that

[X^{H^{r}}, Y^{H^{r}}]_{p} − [X, Y]_{p}^{H^{r}} = Z_{p}^{∗} .

From equation [X^{H}, Y^{H}]_{y} − [X, Y]_{y}^{H} = dΦ_{z}([X^{H^{r}}, Y^{H^{r}}] − [X, Y]^{H^{r}}) we deduce

[X^{H}, Y^{H}]_{y} − [X, Y]_{y}^{H} = dΨ_{z}(Z_{p}^{∗})

To finish the proof of our proposition it is sufficient to verify

A = R̃(X, Y)_{p}(e) (5.13)

This formula follows from Proposition 2.2.□

At the end of this section we prove a formula for the bracket [X^{H}, Y^{(a)}], where a is a nilpotent element of the Weil algebra.

Proposition 5.6 For vector fields X, Y on M and a nilpotent element a ∈ N we have [X^{H}, Y^{(a)}] = (D_{X}J^{r−1}Y)^{(a)} , (5.14)

where D is a covariant derivation of sections of J^{r−1}(T M) with respect to a connection of order r.

PROOF. Since we shall prove (5.14) locally, without lost of generality we can assume that L^{r}M is a trivial bundle.

Let Ψ : L^{r}M × F_{0}(R^{n}) ⟶ F(M) and Φ : L^{r}M × J_{0}^{r−1}(T R^{n}) ⟶ J^{r}(M, R) be the canonical mappings for the associated bundle F(M) and J^{r−1}(T M) respectively. They are defined as follows:

Ψ(j_{0}^{r}γ, z) = F(γ)(z) , Φ(j_{0}^{r}γ, j_{0}^{r−1}W) = j_{γ(0)}^{r−1}(dγ ∘ W ∘ γ^{−1}) .

Let σ : M ⟶ L^{r}M be a section. Let φ_{t} be a 1-parameter group of transformations of X and let φ̂_{t} and φ̃_{t} be the corresponding 1-parameter groups of transformations of X^{H} and X^{H^{r}} respectively.

Let a_{t,x} ∈ L_{n}^{r} such that formula (4.6) holds and let γ_{x} : R^{n} ⟶ M and ξ_{t,x} : R^{n} ⟶ R^{n} be the same mappings that in the proof of Proposition 4.3. Using the introduced notation we have (cf, formula (4.8))

φ̂_{t}(Ψ(σ(x), z)) = Ψ(j_{0}^{r}(γ_{φ_{t}(x)} ∘ ξ_{t,x}), z) = F(γ_{φ_{t}(x)} ∘ ξ_{t,x})(z) . (5.15)

Let us observe that if y ∈ F(M) is a point of F(M) then z = F(γ_{π(y)}^{−1})(y) belongs F_{0}(R^{n}) and

Ψ(σ(π(y)), z) = Ψ(j_{0}^{r}(γ_{π(y)}, F(γ_{π(y)}^{−1})(y)) = y . (5.16)

Now we begin to compute the bracket [X^{H}, Y^{(a)}] where a is nilpotent element. We fix a point y_{0} ∈ F(M) with x_{0} = π(y_{0}) ∈ M. From (5.15) and (5.16) we obtain

[X^{H}, Y^{(a)}]_{y_{0}} = \frac{d}{dt}(dφ̂_{−t}(Y^{(a)}(φ̂_{t}(y_{0}))))_{|t=0}
= \frac{d}{dt}(dφ̂_{−t}(Y^{(a)}(F(γ_{φ_{t}(x_{0})} ∘ ξ_{t,x_{0}} ∘ γ_{x_{0}}^{−1})(y_{0})))_{|t=0}
= \frac{d}{dt}(dφ̂_{−t}(Y^{(a)}(F(A_{t})(y_{0}))))_{|t=0} ,

where A_{t} = γ_{φ_{t}(x_{0})} ∘ ξ_{t,x_{0}} ∘ γ_{x_{0}}^{−1} : M ⟶ M . (5.17) Let (φ̂_{−t})_{F_{φ_{t}(x_{0})}M}) be the restriction of φ̂_{−t} to the fiber F_{φ_{t}(x_{0})}M, since Y^{(a)}(F(A_{t})(y_{0})) is a vertical vector then

dφ̂_{−t}(Y^{(a)}(F(A_{t})(y_{0}))= d((φ̂_{−t})_{F_{φ_{t}(x_{0})}M})(Y^{(a)}(F(A_{t})(y_{0})) .

According to (5.15) and (5.16) we obtain

(φ̂_{−t})_{F_{φ_{t}(x_{0})}M} = F(γ_{φ_{−t}(x_{0})} ∘ ξ_{−t,x_{0}} ∘ γ_{x_{0}}^{−1})

and we have

[X^{H}, Y^{(a)}]_{y_{0}} = \frac{d}{dt}(dF(γ_{φ_{−t}(x_{0})} ∘ ξ_{−t,x_{0}}γ_{x_{0}}^{−1})(Y^{(a)}(F(A_{t})(y_{0}))))_{|t=0}
(5.18)  = \frac{d}{dt}(dF(B_{t})(Y^{(a)}(F(A_{t})(y_{0}))))_{|t=0} ,

where B_{t} = γ_{φ_{−t}(x_{0})} ∘ ξ_{−t,x_{0}} ∘ γ_{x_{0}}^{−1} : M ⟶ M . (5.19)

Using the definition of Y^{(a)} and the identity η_{M} ∘F(dB_{t}) = dF(B_{t})∘η_{M}, where η_{M} : F(T M) ⟶ T (FM) is the natural isomorphism (see [13, 8, 7]) from (5.18) we deduce

[X^{H}, Y^{(a)}]_{y_{0}} = \frac{d}{dt}(dF(B_{t}) ∘ η_{M} ∘ F(m)(a, F(Y ∘ A_{t})(y_{0})))_{|t=0}
= \frac{d}{dt}(η_{M} ∘ F(dB_{t} ∘ m)(a, F(Y ∘ A_{t})(y_{0})))_{|t=0} ,

being m : R × T M ⟶ T M is the multiplication by scalars, m(s, v) = sv. The linearity of dB_{t} means that we have

dB_{t} ∘ m = m ∘ (p_{1}, dB_{t} ∘ p_{2}),

where p_{1} : R × T M ⟶ R and p_{2} : R × T M ⟶ T M are the projections. From this remark we obtain

[X^{H}, Y^{(a)}]_{y_{0}} = \frac{d}{dt}(η_{M} ∘ F(m)(a, F(dB_{t} ∘ Y ∘ A_{t})(y_{0})))_{|t=0}
= \frac{d}{dt}(a · (η_{M} ∘ F(dB_{t} ∘ Y ∘ A_{t})(y_{0})))_{|t=0}
= a · η_{M} (F(\frac{d}{dt}(dB_{t} ∘ Y ∘ A_{t})_{|t=0})(y_{0}))
= Z_{y_{0}}^{(a)} ,

where Z is a vector field on M, which depends of x_{0}, given by

Z = \frac{d}{dt}(dB_{t} ∘ Y ∘ A_{t})_{|t=0} (5.20)

To finish the proof of Proposition 5.6 it is sufficient to verify

(D_{X}j^{r−1}Y)_{x_{0}} = j_{x_{0}}^{r−1}Z , (5.21)

where Z, A_{t}, B_{t} are defined by (5.20), (5.17) and (5.19), respectively.

In order to prove (5.21) we need to study the Crittenden’s mappings asociated to j^{r−1}Y and D_{X}j^{r−1}Y. For a point p ∈ L^{r}M we have

j_{π(p)}^{r−1}Y = Φ(p, j̃^{r−1}Y (p))
(D_{X}j^{r−1}Y)_{π(p)} = Φ(p, D̃_{X}j^{r−1}Y (p)) .

From the first formula and the definition of the canonical mapping Φ for the associated bundle J^{r−1}T M we deduce, for p = j_{0}^{r}γ,

j̃^{r−1}Y (p) = j_{γ(0)}^{r−1}(dγ^{−1} ∘ Y ∘ γ) . (5.22)

On the other hand

D̃_{X}j^{r−1}Y (σ(x_{0})) = X^{H^{r}}(j̃^{r−1}Y)(σ(x_{0}))
= \frac{d}{dt}(j̃^{r−1}Y)(φ̃_{t}(σ(x_{0}))))_{|t=0},

and from (5.22) we have

(D_{X}j^{r−1}Y)_{x_{0}} = Φ(σ(x_{0}), D̃_{X}j^{r−1}Y (σ(x_{0}))) .

Now it is easy to obtain (5.21) by a similar calculus as in the first part of this proof .□

6 Horizontal lifts of tensor fields of type (1, 1) to a product preserving functor

Let Γ be an r-order connection on M. If t is a tensor field of type (1, 1) on M then we define horizontal lift t^{H} with respect to Γ by

\begin{aligned}t^{H}(X^{H}) & = & (tX)^{H} \\ t^{H}(X^{(a)}) & = & (tX)^{(a)} , \end{aligned} (6.1)

where X is a vector field on M and a is a nilpotent element of A = F(R).

According to Proposition 3.4 these formulas determine an unique tensor field of type (1,1) on F(M).

The construction of horizontal lift is natural in the following sense

Proposition 6.1 Let T_{1}^{1}(M) be the space of tensor fields of type (1, 1) on M. If L_{M} denotes the mapping

L_{M} : (Γ, t) ∈ con(M, r) × T_{1}^{1}(M) ⟶ t^{H} ∈ T_{1}^{1}(FM)

then for every mapping φ : M → M′ and for every t ∈ T_{1}^{1}(M), t′ ∈ T_{1}^{1}(M′), Γ ∈ con(M, r), Γ′ ∈ con(M′, r) the following implication

t and t′ are φ − related , Γ and Γ′ are φ − related ⇓ L_{M}(t, Γ) and L_{M′}(t′, Γ′) are F(φ) − related

holds.

From definition of the horizontal lift we deduce the following properties:

Proposition 6.2 The mapping

t ∈ T_{1}^{1}(M) ⟶ t^{H} ∈ T_{1}^{1}(F(M))

is linear over R, where T_{1}^{1}( ) denotes the modul of tensor fields of type (1, 1), it means, for reals α, β and tensors fields t, t′ of type (1, 1) on M we have (αt + βt′)^{H} = αt^{H} + β(t′)^{H} . (6.2)

Furthermore, if t, t′ are two tensor fields of type (1, 1) on M then

(tt′)^{H} = t^{H}t^{′H} . (6.3)

If W (x) is a polynomial with real constant coeficients, then for any tensor field t of type (1, 1) on M we have

W (t^{H}) = (W (t))^{H} . (6.4)

So it t is an almost tangent structure (respectively, an almost complex structure, an f-structure) then so is t^{H}.

Next we study the Nijenhuis tensor N_{t^{H}} of a horizontal lift of a given tensor t of type (1, 1).

Proposition 6.3 Let t be a tensor field field of type (1, 1) on M. If X, Y are vector fields on M and a, b are two nilpotent elements, then

(6.5) N_{t^{H}} (X^{H}, Y^{H}) = (N_{t}(X, Y)^{H} + (t^{2})^{H}(R□(X, Y)) + R□(tX, tY)
−t^{H}(R□(tX, Y) + R□(X, tY)) ,
(6.6) N_{t^{H}} (X^{H}, Y^{(a)}) = (D_{tX}j^{r−1}t · j^{r−1}Y − t · D_{X}j^{r−1}t · j^{r−1}Y)^{(a)}
(6.7) N_{t^{H}} (X^{(a)}, Y^{(b)}) = (N_{t}(X, Y)^{(ab)} ,

where D is the covariant derivation of sections of J^{r−1}(T M ⊗ T^{∗}M) and J^{r−1}(T M), and ” · ” is a natural operations on sections

\underline{J^{r−1}(T M ⊗ T^{∗}M)} × \underline{J^{r−1}(T M)} ⟶ \underline{J^{r−1}(T M)} (6.8)

given by

(Σ · σ)_{x} = j_{x}^{r−1}(t_{x}X_{x}) ,

where Σ(x) = j_{x}^{r−1}t_{x} and σ(x) = j_{x}^{r−1}X_{x}.

PROOF. Formula (6.5) is an immediate consequence of the definition of Nijenhuis tensor and formula (5.12).

To prove formula (6.6) let us observe that if t is a tensor field of type (1,1) on M (a section of T M ⊗ T^{∗}M) and σ is a section of J^{r−1}T M then t · σ = j^{r−1}t · σ. So from (6.1) and (5.14) we obtain

N_{t^{H}} (X^{H}, Y^{(a)}) = (D_{tX}j^{r−1}(tY))^{(a)} − t^{H}(D_{tX}j^{r−1}Y)^{(a)}
+ D_{X}j^{r−1}(tY))^{(a)}) + (t^{2})^{H}((D_{X}j^{r−1}Y)^{(a)})
= (D_{tX}j^{r−1}(tY) − t · D_{tX}j^{r−1}(Y) + t · D_{X}j^{r−1}(tY)
+ t^{2} · D_{X}j^{r−1}(Y))^{(a)} .

Let us remark that j^{r−1}(tY) = j^{r−1}t · j^{r−1}Y. Now from the bilinearity of (6.8) we have

D_{tX}j^{r−1}(tY) = D_{tX}j^{r−1}t · j^{r−1}Y + j^{r−1}t · D_{tX}j^{r−1}Y
= D_{tX}j^{r−1}t · j^{r−1}Y + t · D_{tX}j^{r−1}Y ,
D_{X}j^{r−1}(tY) = D_{X}j^{r−1}t · j^{r−1}Y + j^{r−1}t · D_{X}j^{r−1}Y
= D_{X}j^{r−1}t · j^{r−1}Y + t · D_{X}j^{r−1}Y .

Using these equations we obtain formula (6.6) . Formula (6.7) follows immediately from Proposition 3.2. □

Now, we shall study the integrability of lifted geometric structures from M to F(M).

Proposition 6.4 Let t be a complex structure M such that for all vector fields X on M we have D_{X}j^{r−1}t = 0 , (6.9)

where D is a covariant derivation with respect to a connection of order r on M. The almost complex structure t^{H} on F(M) is integrable if and only if for all vector fields X, Y on M we have

R(tX, tY) = R(X, Y) , (6.10)

where R( , ) is the curvature transformation of sections of J^{r}(M, R).

PROOF. Since N_{t} = 0 from (6.9) and Proposition 6.3 we obtain

N_{t^{H}} (X^{H}, Y^{H}) = −R□(X, Y) + R□(tX, tY)

−t^{H}(R□(tX, Y) + R□(X, tY)) ,

N_{t^{H}} (X^{H}, Y^{(a)}) = N_{t^{H}} (X^{(a)}, Y^{(b)}) = 0 .

Since condition (6.10) is equivalent to R(tX, Y) = −R(X, tY), N_{t^{H}} = 0 if and only if formula (6.10) holds.□

Analogously we can verify as immediate consequence of Proposition 6.3 the following propositions.

Proposition 6.5 Let t be a product structure M such that for all vector fields X on M we have D_{X}j^{r−1}t = 0 , (6.11)

where D is a covariant derivation with respect to a connection of order r on M. The almost product structure t^{H} on F(M) is integrable if and only if for all vector fields X, Y on M we have

R(tX, tY) = −R(X, Y) , (6.12)

where R( , ) is the curvature transformation of sections of J^{r}(M, R).

Proposition 6.6 Let t be a tangent structure M such that for all vector fields X on M we have D_{X}j^{r−1}t = 0 , (6.13)

where D is a covariant derivation with respect to a connection of order r on M. If for all vector fields X, Y on M we have

R(tX, Y) = 0 , (6.14)

where R( , ) is the curvature transformation of sections of J^{r}(M, R), then the almost tangent structure t^{H} on F(M) is integrable.

7 Horizontal lift of linear connections to a product preserving functor

Proposition 7.1 Let Γ ∈ con(M, r) and ∇ be a linear connection on M. Then there exists one and only one linear connection ∇^{H} on F(M) such that

(7.1)  ∇_{X^{H}}^{H} Y^{H} = (∇_{X}Y)^{H}
(7.2)  ∇_{X^{H}}^{H} Y^{(a)} = [X^{H}, Y^{(a)}]
(7.3)  ∇_{X^{(a)}}^{H}Y^{H} = 0
(7.4)  ∇_{X^{(a)}}^{H}Y^{(b)} = (∇_{X}Y)^{(ab)}

PROOF. It is similar to a proof given in [9].□

The unique linear connection ∇^{H} on F(M) is called the horizontal lift of ∇ with respect to Γ.

Let us remark that in the above proposition we do not suppose any relationship between the linear connection ∇ and the connection Γ.

If the product preserving functor F is the tangent bundle of order r then ∇^{H} coincides with the connection defined in [9]. If F(M) is the tangent bundle and connections ∇ and Γ coincide then ∇^{H} is the connection defined by K. Yano and E. Ishihara in [19] (see also [20]).

Directly from definitions we can obtain the torsion of ∇^{H}.

Proposition 7.2 Let T and T̃ be the torsion tensor of ∇ and ∇^{H} respectively. If X, Y are vector fields on M and a, b are nilpotent element of Weil algebra, then

T̃(X^{H}, Y^{H}) = (T (X, Y))^{H} − R□(X, Y)
T̃(X^{H}, Y^{(a)}) = 0
T̃(X^{(a)}, Y^{(b)}) = (T (X, Y))^{(ab)}

This proposition implies immediately

Corollary 7.3 The horizontal lift ∇^{H} is a torsionless connection on F(M) if and only if a connection ∇ is torsionless on M and curvature transformation of the connection of order R vanishes.

The curvature tensor of ∇^{H} has not been computed because there is not a good expression for [R(X, Y)^{□}, Y^{(a)}]. Nevertheless, it can be easily checked that if ∇ has neither torsion nor curvature and the curvature transformation of Γ is null, then ∇^{H} is also torsionless and curvatureless.

If F(M) is the tangent bundle of order r then Proposition 7.3 has been proved in [9]. If F(M) is the tangent bundle and connections ∇ and Γ coincide this proposition has been proved by K. Yano and S. Ishihara in [19] (see also [20]).

The horizontal lifts of tensors and connections to the principal bundles LM and L^{2}M considered in [1], as well as their properties, arise by restriction of our horizontal lifts to T^{n,1}M and T^{n,2}M.

Proposition 7.4 Let t be a tensor field of type (1,1) on M, ∇ be a linear connection on M and Γ be a connection of order r on M. If X, Y are vector fields on M and a, b are nilpotent elements of Weil algebra then

(∇_{X^{H}} t^{H})(Y^{H}) = ((∇_{X}t)(Y))^{H}
(∇_{X^{H}} t^{H})(Y^{(a)}) = (D_{X}(j^{r−1}t) · j^{r−1}Y))^{(a)}
(∇_{X^{(a)}}t^{H})(Y^{H}) = 0
(∇_{X^{(a)}}t^{H})(Y^{(b)}) = ((∇_{X}t)(Y))^{(ab)}

PROOF. Using formulas (7.1), (7.2), (7.3), (7.4) and (5.12) we can verify the proposition by a standard evaluation.□

From Proposition 7.4 we deduce immediately

Corollary 7.5 Let t be a tensor field of type (1,1) on M, ∇ be a linear connection on M and Γ be a connection of order r on M. ∇^{H} is a t^{H}- connection on F(M) if and only if ∇ is a t-connection on M and D_{X}j^{r−1}t = 0 for all vector fields X on M.

If F(M) is the tangent bundle and connections ∇ and Γ coincide then Proposition 7.5 has be proved by K. Yano and S. Ishihara [19], [20].

References

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Author’s address:

Jacek Gancarzewicz, Instytut Matematyki UJ, ul. Reymonta 4, 30-059 Kraków, POLAND, e-mail: gancarze@im.uj.edu.pl

Modesto Salgado, Departamento de Xeometria e Topoloxia, Facultade de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela, SPAIN, e-mail: modesto@zmat.usc.es

Noureddine Rahmani Institute de Mathematiques Appliques et Informatique, USTO, Oran, ALGERIA,