Caracterizations of the Nonlinear Connection in the Higher Order Geometry

Ioan Bucataru

Abstract

In the paper [8] a nonlinear connection on k-osculator bundle is characterised by a system of functions defined on each domain of local chart, which verify a special formula.

Starting with this result, to a nonlinear connection on the k-osculator bundle was associated a special map, called connection map [2].

The aim of this paper is to present the notion of connection map, independently of the choice of a nonlinear connection. The kernel of the connection map is a nonlinear connection. In this way we obtain a characterization for the nonlinear connection using only the k-tangent structure on the k-osculator bundle.

In the last part of this paper we present the notion of horizontal lift, independently of the choice of a nonlinear connection and connection map. Using this map we obtain a characterization for a connection map and a nonlinear connection. The nonlinear connection appears as the image of the horizontal lift. The connection map is defined using the inverse map of the vertical lift.

AMS subject classification: 53C60 Key words: nonlinear connection, connection map, horizontal lift

1 Introduction

Let M be a real, smooth manifold of dimension n and (Osc^{k}M, π^{k}, M) its k-osculator bundle. Then Osc^{k}M is a real, smooth manifold of dimension n(k + 1). We set E = Osc^{k}M.

Let (x^{i}) be the local coordinates in a local chart U ⊂ M. The local coordinates on (π^{k})^{−1}(U) ⊂ Osc^{k}M will be denoted by (x^{i}, y^{(1)i}, …, y^{(k)i}).

A change of coordinates (x, y^{(1)}, ..., y^{(k)}) → (x̃, y^{(1)}, …, y^{(k)}) on E is given by:

Balkan Journal of Geometry and Its Applications, Vol.2, No.2, 1997, pp. 13-22 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

(1.1)  \{ \begin{matrix}x^{i} = x^{i}(x^{1}, x^{2}, …, x^{n}); rang ∥\frac{∂x^{i}}{∂x^{j}} ∥ = n \\ y^{(1)i} = \frac{∂x^{i}}{∂x^{j}} y^{(1)j} \\ 2y^{(2)i} = \frac{∂y^{(1)i}}{∂x^{j}} y^{(1)j} + 2\frac{∂y^{(1)i}}{∂y^{(1)j}} y^{(2)j} \\ … \\ ky^{(k)i} = \frac{∂y^{(k−1)i}}{∂x^{j}} y^{(1)j} + 2\frac{∂y^{(k−1)i}}{∂y^{(1)j}} y^{(2)j} + … + k \frac{∂y^{(k−1)i}}{∂y^{(k−1)j}} y^{(k)j}. \end{matrix}

Under a change of coordinates (1.1) on E, for each u ∈ E, the natural basis

\{\frac{∂}{∂x^{i}} |_{u}, \frac{∂}{∂y^{(1)i}} |_{u}, …, \frac{∂}{∂y^{(k)i}} |_{u}\} changes as follows

(1.2) \{ \begin{matrix}\frac{∂}{∂x^{i}} |_{u}= \frac{∂x^{j}}{∂x^{i}} (u)\frac{∂}{∂x^{j}} |_{u} +\frac{∂y^{(1)j}}{∂x^{i}} (u)\frac{∂}{∂y^{(1)j}} |_{u} + … + \frac{∂y^{(k)j}}{∂x^{i}} (u)\frac{∂}{∂y^{(k)j}} |_{u} \\ \frac{∂}{∂y^{(1)i}} |_{u}= \frac{∂y^{(1)j}}{∂y^{(1)i}} (u)\frac{∂}{∂y^{(1)j}} |_{u} + … + \frac{∂y^{(k)j}}{∂y^{(1)i}} (u)\frac{∂}{∂y^{(k)j}} |_{u} \\ ⋯ \\ \frac{∂}{∂y^{(k)i}} |_{u}= \frac{∂y^{(k)j}}{∂y^{(k)i}} (u)\frac{∂}{∂y^{(k)j}} |_{u} . \end{matrix}

For α ∈ {1, 2, …, k − 1} we denote π_{α}^{k} : E → Osc^{α}M the canonical submersion which is expressed in the local chart as follows

π_{α}^{k} : (x, y^{(1)}, …, y^{(k)}) ↦ (x, y^{(1)}, …, y^{(α)}).

As (π^{k})_{∗} : (T E, τ_{E}, E) → (T M, τ, M) is a π^{k} epimorphism of vector bundles, it results that its kernel is a vector subbundle of the bundle (T E, τ_{E}, E). This will be denoted by V_{1}E and will be called the vertical subbundle of the T E. The fibres of V_{1}E determine an integrable distribution V_{1} : u ∈ E ↦ V_{1}(u) ⊂ T_{u}E which has the dimension kn, called vertical distribution.

In the same manner, for each submersion π_{α}^{k} : E → Osc^{α}M we obtain a vector subbundle of T E denoted by V_{α+1}E = Ker(π_{α}^{k})_{∗}. The fibres of V_{α+1}E determine an integrable distribution V_{α+1} : u ∈ E ↦ V_{α+1}(u) ⊂ T_{u}E, of dimension (k − α)n.

In this way we obtain k-vertical distributions V_{1}, V_{2}, …, V_{k} of dimensions kn, (k − 1)n, …, n respectively, such that ∀u ∈ E, V_{k}(u) ⊂ V_{k−1}(u) ⊂ ⋯ ⊂ V_{1}(u).

For each u ∈ E we consider the linear map J_{u} : T_{u}E → T_{u}E defined on natural basis by (1.3)

J_{u} (\frac{∂}{∂x^{i}} |_{u}) = \frac{∂}{∂y^{(1)i}} |_{u}, …, J_{u} (\frac{∂}{∂y^{(k−1)i}} |_{u}) = \frac{∂}{∂y^{(k)i}} |_{u}, J_{u} (\frac{∂}{∂y^{(k)i}} |_{u}) = 0

and extended by linearity. Proposition 1.1.

1. For each u ∈ E the map J_{u} is well defined.

2. For α ∈ {1, 2, …, k − 1} we have

(1.4)  \{ \begin{matrix}J_{u}^{k−α}V_{k−α}(u) = V_{α}(u), ∀u ∈ E \\ J_{u}^{k+1} = 0 \end{matrix} .

The maps J_{u}, u ∈ E determine a morphism of vector bundles J : (T E, τ_{E}, E) → (T E, τ_{E}, E) and an F(E) linear map J : χ(E) → χ(E). The map J is called the k–tangent structure of the k–osculator bundle.

2 The nonlinear connection associated to a connection map

We denote by (T M^{(k)}, τ^{(k)}, M) the Whitney sum of the tangent bundle (T M, τ, M) on itself of k times. Definition 2.1. We call connection map on the k-osculator bundle E a π^{k} morphism of vector bundles

K = (K^{(1)}, K^{(2)}, …, K^{(k)}) : (T E, τ_{E}, E) → (T M^{(k)}, τ^{(k)}, M)

which satisfies

(2.1)  \{ \begin{matrix}K^{(k)} ∘J^{α} = K^{(k−α)}, ∀α = \overline{1, k − 1} \\ K^{(k)} ∘J^{k} = (π^{k})_{∗} \end{matrix} .

Proposition 2.1. For a connection map K = (K^{(1)}, K^{(2)}, …, K^{(k)}) we have the following relations

(2.2)  \{ \begin{matrix}K^{(α)}= K^{(α+1)} ∘J, ∀α = \overline{1, k − 1} \\ (π^{k})_{∗} =K^{(α)} ∘J^{α} ∀α = \overline{1, k} \end{matrix} .

Proof.

K^{(α)}=K^{(k)} ∘J^{k−α} =K^{(k)} ∘J^{k−α−1} ∘ J = K^{(α+1)} ∘J

(π^{k})_{∗} =K^{(k)} ∘J^{k} =K^{(k)} ∘J^{k−1} ∘ J =K^{(1)} ∘J

Remark 2.1. The kernel of the connection map N = KerK is a vector subbundle of the bundle (T E, τ_{E}, E). Its fibres determine a distribution N : u ∈ E ↦ N(u) ⊂ E_{u} of dimension n, called the horizontal distribution associated to the connection map K.

Next, we shall prove that a connection map determines a nonlinear connection on k–osculator bundle.

For each u ∈ E, the map K_{u} : T_{u}E → \underbrace{T_{π^{k}(u)}M × ⋯ × T_{π^{k}(u)}M}_{k times} is linear. We

denote by M_{(1)} _{j}^{i}(u), …, M_{(k)} _{j}^{i}(u) the coordinate functions, defined on every domain of local charts, for the vectors K_{u}^{(1)} \frac{∂}{∂x^{j}} |_{u}, …, and K_{u}^{(k)} \frac{∂}{∂x^{j}} |_{u} respectively, in the natural

basis \frac{∂}{∂x^{i}} |_{π^{k}(u)} of T_{π^{k}(u)}M. Therefore

(2.3)  K_{u} \frac{∂}{∂x^{j}} |_{u}= (M_{(1)} _{j}^{i}(u)\frac{∂}{∂x^{i}} |_{π^{k}(u)}, …, M_{(k)} _{j}^{i}(u)\frac{∂}{∂x^{i}} |_{π^{k}(u)}).

Taking account of Proposition 2.1 and formula (2.3), we obtain the following formulae

(2.4) \{ \begin{matrix}K_{u} \frac{∂}{∂y^{(1)j}} |_{u}= (\frac{∂}{∂x^{i}} |_{π^{k}(u)}, M_{(1)} _{j}^{i}(u)\frac{∂}{∂x^{i}} |_{π^{k}(u)}, …, M_{(k−1)}_{j}^{i}(u)\frac{∂}{∂x^{i}} |_{π^{k}(u)}) \\ K_{u} \frac{∂}{∂y^{(2)j}} |_{u}= (0, \frac{∂}{∂x^{i}} |_{π^{k}(u)}, M_{(1)} _{j}^{i}(u)\frac{∂}{∂x^{i}} |_{π^{k}(u)}, …, M_{(k−2)}_{j}^{i}(u)\frac{∂}{∂x^{i}} |_{π^{k}(u)}) \\ ⋯ \\ K_{u} \frac{∂}{∂y^{(k−1)j}} |_{u}= (0, …, 0, \frac{∂}{∂x^{i}} |_{π^{k}(u)}, M_{(1)} _{j}^{i}(u)\frac{∂}{∂x^{i}} |_{π^{k}(u)}) \\ K_{u} \frac{∂}{∂y^{(k)j}} |_{u}= (0, …, 0, \frac{∂}{∂x^{i}} |_{π^{k}(u)}). \end{matrix}

Theorem 2.1. Under a change of coordinates (1.1) on E, the set of functions (M_{(α)} _{j}^{i})_{α=\overline{1,k}} is changing according to the following rules

(2.5) \{ \begin{matrix}M_{(1)} _{j}^{m} \frac{∂x^{i}}{∂x^{m}} = ˜ M_{(1)} _{m}^{i} \frac{∂x^{m}}{∂x^{j}} + \frac{∂y^{(1)i}}{∂x^{j}} \\ M_{(2)} _{j}^{m} \frac{∂x^{i}}{∂x^{m}} = ˜ M_{(2)} _{m}^{i} \frac{∂x^{m}}{∂x^{j}} + ˜ M_{(1)} _{m}^{i} \frac{∂y^{(1)m}}{∂x^{j}} + \frac{∂y^{(2)i}}{∂x^{j}} \\ ⋯ \\ M_{(k)} _{j}^{m} \frac{∂x^{i}}{∂x^{m}} = ˜ M_{(k)} _{m}^{i} \frac{∂x^{m}}{∂x^{j}} + ˜ M_{(k−1)}_{m}^{i} \frac{∂y^{(1)m}}{∂x^{j}} + … + ˜ M_{(1)} _{m}^{i} \frac{∂y^{(k−1)m}}{∂x^{j}} + \frac{∂y^{(k)i}}{∂x^{j}} . \end{matrix}

Proof. For (U, ϕ = (x^{i}, y^{(1)i}, …, y^{(k)i})) and (V, ψ = (x^{i}, y^{(1)i}, …, y^{(k)i})) two local charts in u ∈ U ∩ V, we have

K_{u} \frac{∂}{∂x^{j}} |_{u}= (˜M_{(1)} _{j}^{i}(u)\frac{∂}{∂x^{i}} |_{π^{k}(u)}, …, ˜ M_{(k)} _{j}^{i}(u)\frac{∂}{∂x^{i}} |_{π^{k}(u)}) = K_{u} \frac{∂}{∂x^{j}} |_{u} .

According to (1.2),

\begin{matrix}K_{u} \frac{∂}{∂x^{i}} |_{u}= K_{u}(\frac{∂x^{m}}{∂x^{i}} (u)\frac{∂}{∂x^{m}} |_{u} +\frac{∂y^{(1)m}}{∂x^{i}} (u)\frac{∂}{∂y^{(1)m}} |_{u} + … + \\ +\frac{∂y^{(k)m}}{∂x^{i}} (u)\frac{∂}{∂y^{(k)m}} |_{u}) = \frac{∂x^{m}}{∂x^{i}} (u)K_{u} \frac{∂}{∂x^{m}} |_{u} +\frac{∂y^{(1)m}}{∂x^{i}} (u)K_{u} \frac{∂}{∂y^{(1)m}} |_{u} + … + \\ +\frac{∂y^{(k)m}}{∂x^{i}} (u)K_{u} \frac{∂}{∂y^{(k)m}} |_{u}= \frac{∂x^{m}}{∂x^{i}} (M_{(1)} _{m}^{j}(u)\frac{∂}{∂x^{j}} |_{π^{k}(u)}, …, M_{(k)} _{m}^{j}(u)\frac{∂}{∂x^{j}} |_{π^{k}(u)})+ \\ +\frac{∂y^{(1)m}}{∂x^{i}} (\frac{∂}{∂x^{m}} |_{π^{k}(u)}, M_{(1)} _{m}^{j}(u)\frac{∂}{∂x^{j}} |_{π^{k}(u)}, …, M_{(k−1)}_{m}^{j}(u)\frac{∂}{∂x^{j}} |_{π^{k}(u)}) + … + \\ +\frac{∂y^{(k−1)m}}{∂x^{i}} (0, …, 0, \frac{∂}{∂x^{m}} |_{π^{k}(u)}, M_{(1)} _{m}^{j}(u)\frac{∂}{∂x^{j}} |_{π^{k}(u)}) + \frac{∂y^{(k)m}}{∂x^{i}} (0, …, 0, \frac{∂}{∂x^{m}} |_{π^{k}(u)}). \end{matrix} .

On this way we obtain

\{ \begin{matrix}˜M_{(1)} _{i}^{j}(u)\frac{∂}{∂x^{j}} |_{π^{k}(u)}= \frac{∂x^{m}}{∂x^{i}} M_{(1)} _{m}^{j}(u)\frac{∂}{∂x^{j}} |_{π^{k}(u)} +\frac{∂y^{(1)j}}{∂x^{i}} \frac{∂}{∂x^{j}} |_{(π^{k}(u)} \\ ⋯ \\ ˜M_{(k)} _{i}^{j}(u)\frac{∂}{∂x^{j}} |_{π^{k}(u)}= \frac{∂x^{m}}{∂x^{i}} M_{(k)} _{m}^{j}(u)\frac{∂}{∂x^{j}} |_{π^{k}(u)} +\frac{∂y^{(1)m}}{∂x^{i}} M_{(k−1)}_{m}^{j}(u)\frac{∂}{∂x^{j}} |_{π^{k}(u)})+ \\ + … + \frac{∂y^{(k)j}}{∂x^{i}} \frac{∂}{∂x^{j}} |_{π^{k}(u)} . \end{matrix}

and using \frac{∂}{∂x^{j}} |_{π^{k}(u)}= \frac{∂x^{s}}{∂x^{j}} \frac{∂}{∂x^{s}} |_{π^{k}(u)} it results (2.5).

Next, on every domain of local chart on E we consider the set of functions

(N_{(1)} _{j}^{i}, …, N_{(k)}_{j}^{i}) defined by

(2.6)  \{ \begin{matrix}N_{(1)} _{j}^{i} = M_{(1)} _{j}^{i} \\ N_{(2)} _{j}^{i} = M_{(2)} _{j}^{i} − N_{(1)} _{m}^{i}M_{(1)} _{j}^{m} \\ … \\ N_{(k)}_{j}^{i} = M_{(k)} _{j}^{i} − N_{(k−1)}_{m}^{i}M_{(1)} _{j}^{m} − … − N_{(1)} _{m}^{i} M_{(k−1)}_{j}^{m}. \end{matrix}

We use the following notations

(2.7)  \{ \begin{matrix}\frac{δ}{δx^{i}} |_{u}= \frac{∂}{∂x^{i}} |_{u} −N_{(1)} _{i}^{j}(u)\frac{∂}{∂y^{(1)j}} |_{u} − … − N_{(k)}_{i}^{j}(u)\frac{∂}{∂y^{(k)j}} |_{u} \\ \frac{δ}{δy^{(1)i}} |_{u}= J_{u} (\frac{δ}{δx^{i}} |_{u}) , …, \frac{δ}{δy^{(k−1)i}} |_{u}= J_{u} (\frac{δ}{δy^{(k−2)i}} |_{u}) . \end{matrix}

Theorem 2.2. The vector fields

\frac{δ}{δx^{i}} , \frac{δ}{δy^{(1)i}} , ..., \frac{δ}{δy^{(k−1)i}} , \frac{∂}{∂y^{(k)i}}

are d-vector fields, that is under a change of coordinates (1.1) these are changing by the following rules

(2.8)  \{ \begin{matrix}\frac{δ}{δx̃^{i}} = \frac{∂x^{j}}{∂x̃^{i}} \frac{δ}{δx^{j}} , \\ \frac{δ}{δỹ^{(α)i}} = \frac{∂x^{j}}{∂x̃^{i}} \frac{δ}{δy^{(α)j}} (α = \overline{1, k − 1}), \\ \frac{∂}{∂ỹ^{(k)i}} = \frac{∂x^{j}}{∂x̃^{i}} \frac{∂}{∂y^{(k)j}} . \end{matrix}

Proof. Under a change of coordinates on E, the set of functions (N_{(1)} _{j}^{i}, …, N_{(k)}_{j}^{i}) changes as follows

(2.9)  \{ \begin{matrix}˜N_{(1)} _{m}^{i} \frac{∂x^{m}}{∂x^{j}} = N_{(1)} _{j}^{m} \frac{∂x^{i}}{∂x^{m}} − \frac{∂y^{(1)i}}{∂x^{j}} \\ ˜N_{(2)} _{m}^{i} \frac{∂x^{m}}{∂x^{j}} = N_{(2)} _{j}^{m} \frac{∂x^{i}}{∂x^{m}} + N_{(1)} _{j}^{m} \frac{∂y^{(1)i}}{∂x^{m}} − \frac{∂y^{(2)i}}{∂x^{j}} \\ ⋯ \\ ˜N_{(k)}_{m}^{i} \frac{∂x^{m}}{∂x^{j}} = N_{(k)}_{j}^{m} \frac{∂x^{i}}{∂x^{m}} + N_{(k−1)}_{j}^{m} \frac{∂y^{(1)i}}{∂x^{m}} + … + N_{(1)} _{j}^{m} \frac{∂y^{(k−1)i}}{∂x^{m}} − \frac{∂y^{(k)i}}{∂x^{j}} . \end{matrix}

Using (2.9) and (1.2) it results that under a change of coordinates (1.1) on E, (2.8) is

true. The vectors \frac{δ}{δx^{i}} |_{u}, \frac{δ}{δy^{(1)i}} |_{u}, ..., \frac{δ}{δy^{(k−1)i}} |_{u}, \frac{∂}{∂y^{(k)i}} |_{u} are linearly independent

on T_{u}E, ∀u ∈ E. Proposition 2.2. We have the following formulae

(2.10)  \{ \begin{matrix}(π^{k})_{∗,u} (\frac{δ}{δx^{i}} |_{u}) = \frac{∂}{∂x^{i}} |_{π^{k}(u)} \\ K^{(α)}_{u} (\frac{δ}{δy^{(α)i}} |_{u}) = \frac{∂}{∂x^{i}} |_{π^{k}(u)} ∀α = \overline{1, k − 1} \\ K^{(k)}_{u} (\frac{∂}{∂y^{(k)i}} |_{u}) = \frac{∂}{∂x^{i}} |_{π^{k}(u)} . \end{matrix}

Proof.

(π^{k})_{∗,u} (\frac{δ}{δx^{i}} |_{u}) = (π^{k})_{∗,u} (\frac{∂}{∂x^{i}} |_{u} −N_{(1)} _{i}^{j}(u)\frac{∂}{∂y^{(1)j}} |_{u} − … − N_{(k)}_{i}^{j}(u)\frac{∂}{∂y^{(k)j}} |_{u}) =

= (π^{k})_{∗,u} (\frac{∂}{∂x^{i}} |_{u}) = \frac{∂}{∂x^{i}} |_{π^{k}(u)} .

The other formulae are proved by using (2.2) and (2.7).

Theorem 2.3. For a connection map K = (K^{(1)}, K^{(2)}, …, K^{(k)}) : (T E, τ_{E}, E) → (T M^{(k)}, τ^{(k)}, M),

its kernel N = KerK is a nonlinear connection on k-osculator bundle. Proof. To prove this theorem it is sufficient to show the following decomposition in the Whitney sum

(2.11)  T E = N ⊕ V_{1}.

Firstly, we prove that \{\frac{δ}{δx^{i}} |_{u}\}_{i=\overline{1,n}} is a basis for KerK_{u} = N(u). Since \{\frac{δ}{δx^{i}} |_{u}\}_{i=\overline{1,n}}

are n linearly independent vectors and the dimension of N(u) is n, we need to prove

that K_{u} (\frac{δ}{δx^{i}} |_{u}) = 0. We have the following sequence of equalities

K_{u}(\frac{δ}{δx^{i}} |_{u}) = K_{u}(\frac{∂}{∂x^{i}} |_{u} −N_{(1)} _{i}^{j}(u)\frac{∂}{∂y^{(1)j}} |_{u} − … − N_{(1)} _{i}^{j}(u)\frac{∂}{∂y^{(k)j}} |_{u}) =

= K_{u}(\frac{∂}{∂x^{i}} |_{u}) − N_{(1)} _{i}^{j}(u)K_{u}(\frac{∂}{∂y^{(1)j}} |_{u}) − … − N_{(k)}_{i}^{j}(u)K_{u}(\frac{∂}{∂y^{(k)j}} |_{u}) =

(M_{(1)} _{j}^{i}(u)\frac{∂}{∂x^{i}} |_{π^{k}(u)}, …, M_{(k)} _{j}^{i}(u)\frac{∂}{∂x^{i}} |_{π^{k}(u)}) − N_{(1)} _{j}^{i}(u)(\frac{∂}{∂x^{i}} |_{π^{k}(u)}, M_{(1)} _{j}^{m}(u)\frac{∂}{∂x^{m}} |_{π^{k}(u)},

, …, M_{(k−1)}_{j}^{m}(u)\frac{∂}{∂x^{m}} |_{π^{k}(u)}) − N_{(2)} _{j}^{i}(u)(0, \frac{∂}{∂x^{i}} |_{π^{k}(u)}, M_{(1)} _{j}^{m}(u)\frac{∂}{∂x^{m}} |_{π^{k}(u)}, …,

M_{(k−2)}_{j}^{i}(u)\frac{∂}{∂x^{i}} |_{π^{k}(u)}) − ⋯ − N_{(k−1)}_{j}^{i}(u)(0, …, 0, \frac{∂}{∂x^{i}} |_{π^{k}(u)}, M_{(1)} _{j}^{m}(u)\frac{∂}{∂x^{m}} |_{π^{k}(u)})−

−N_{(k)}_{j}^{i}(u)(0, …, 0, \frac{∂}{∂x^{i}} |_{π^{k}(u)}).

Using (2.6) one obtains K_{u} (\frac{δ}{δx^{i}} |_{u}) = 0. Finally, we have to prove that ∀u ∈ E,

N(u) ∩ V_{1}(u) = {0}. Let X_{u} ∈ N(u) ∩ V_{1}(u). Because X_{u} ∈ N(u) it results K_{u}X_{u} = 0 and because X_{u} ∈ V_{1}(u) we have (π^{k})_{∗,u}X_{u} = 0. If X_{u} is expressed in the basis (2.7) by

X_{u} =X^{i}^{(0)} \frac{δ}{δx^{i}} + X^{i}^{(1)} \frac{δ}{δy^{(1)i}} + … + X^{i}^{(k)} \frac{∂}{∂y^{(k)i}}

it follows that

K_{u}X_{u} = (X^{i}^{(1)} \frac{∂}{∂x^{i}} , X^{i}^{(2)} \frac{∂}{∂x^{i}} , …, X^{i}^{(k)} \frac{∂}{∂x^{i}}) = 0

and

(π^{k})_{∗,u}X_{u} =X^{i}^{(0)} \frac{∂}{∂x^{i}} = 0 ⟹X^{i}^{(0)}=X^{i}^{(1)}= ⋯ =X^{i}^{(k)}= 0.

We call the functions (N_{(1)} _{j}^{i}, …, N_{(k)}_{j}^{i}) the coefficients of the nonlinear connection N.

Next, we prove the existence of connection map. Theorem 2.4. If the manifold M is smooth, then there exists a connection map on the k-osculator bundle of the manifold M. Proof. In the paper [2] we have proved that every nonlinear connection N on the k–osculator bundle determines a connection map. Because on Osc^{k}M there exist nonlinear connections, the proof is finished.

We denote N_{0} = N, N_{1} = J(N_{0}), ⋯, N_{k−1} = J(N_{k−2}). The following decomposition in the Whitney sum is true

(2.12)  T E = N_{0} ⊕ N_{1} ⊕ ⋯ ⊕ N_{k−1} ⊕ V_{k}.

The distributions N_{α} : u ∈ E ↦ N_{α}(u) ⊂ T_{u}E are of dimensions n. Generally these are not integrable.

Concluding, the existence of the nonlinear connection N on the k-osculator bundle E is characterized by a special π^{k} morphism of vector bundles K such that N = Ker K.

3 Characterizations of the connection map with the lifts.

Let (V_{k}E, τ_{E} |_{V_{k}E}, E) be a vector bundle over E, whose fibres are V_{k}(u), u ∈ E. For each u ∈ E, we denote by (l_{v_{k}})_{π^{k}(u),u} : T_{π^{k}(u)}M → V_{k}(u) the linear map

defined by: (l_{v_{k}})_{π^{k}(u),u} (\frac{∂}{∂x^{i}} |_{π^{k}(u)}) = \frac{∂}{∂y^{(k)i}} |_{u} and extended by linearity.

The map (l_{v_{k}})_{π^{k}(u),u} is a linear isomorphism for ∀u ∈ E. It is called the vertical lift. Definition 3.1. The horizontal lift is defined as a linear map (l_{h})_{π^{k}(u),u} : T_{π^{k}(u)}M → T_{u}E which satisfies

(3.1)  (l_{v_{k}})_{π^{k}(u),u} = J_{u}^{k} ∘ (l_{h})_{π^{k}(u),u}.

We use the notation N(u) = (l_{h})_{π^{k}(u),u}(T_{π^{k}(u)}M).

Remark. The map (l_{h})_{π^{k}(u),u} : T_{π^{k}(u)}M → N(u) is a linear isomorphism of vector spaces. Proposition 3.1. Every connection map K on the k–osculator bundle E determines a horizontal lift. Proof. The map π_{∗,u}^{k} |_{Ker K_{u}}: Ker K_{u} → T_{π^{k}(u)}M is an isomorphism of vector spaces. We denote by (l_{h})_{π^{k}(u),u} : T_{π^{k}(u)}M → Ker K_{u} its inverse map. We have

π_{∗,u}^{k} ∘ (l_{h})_{π^{k}(u),u} = id_{T_{π^{k}(u)}M}. Firstly, we prove K_{u}^{(k)} ∘(l_{v_{k}})_{π^{k}(u),u} = id_{T_{π^{k}(u)}M}. Using

that (l_{v_{k}})_{π^{k}(u),u} (\frac{∂}{∂x^{i}} |_{π^{k}(u)}) = \frac{∂}{∂y^{(k)i}} |_{u} and K_{u}^{(k)} (\frac{∂}{∂y^{(k)i}} |_{u}) = \frac{∂}{∂x^{i}} |_{π^{k}(u)} we

obtain the previous formula. The proof of proposition is finished by the following

sequence of implications: π_{∗,u}^{k} =K_{u}^{(k)} ∘J_{u}^{k} ⇒K_{u}^{(k)} ∘J_{u}^{k} ∘ (l_{h})_{π^{k}(u),u} =K_{u}^{(k)} ∘(l_{v_{k}})_{π^{k}(u),u} ⇒

the map l_{h} verify (3.1). Proposition 3.2. Every horizontal lift determines a connection map on k-osculator bundle. Proof. Let (l_{h})_{π^{k}(u),u} : T_{π^{k}(u)}M → T_{u}E be a map which satisfy (3.1).

Let K_{u}^{(k)}: V_{k}(u) → T_{π^{k}(u)} be the inverse map of the vertical lift and K_{u} = (K_{u}^{(k)}

∘J_{u}^{k−1}, … K_{u}^{(k)} ∘J_{u}, K_{u}^{(k)}). For proving that the map K is a connection map it is sufficient

to show that K_{u}^{(k)} ∘J_{u}^{k} = π_{∗,u}^{k}. From (3.1), compound at left by K_{u}^{(k)} we have K_{u}^{(k)} ∘J_{u}^{k} ∘

(l_{h})_{π^{k}(u),u} =K_{u}^{(k)} ∘(l_{v_{k}})_{π^{k}(u),u} = id_{T_{π^{k}(u)}M}. Since (l_{h})_{π^{k}(u),u} is a linear isomorphism it

results that K_{u}^{(k)} ∘J_{u}^{k} is its inverse and K_{u}^{(k)} ∘J_{u}^{k} = π_{∗,u}^{k}.

For α ∈ {1, …, k − 1} and u ∈ E we denote (l_{v_{α}})_{π^{k}(u),u} : T_{π^{k}(u)}M → T_{u}E the map defined by (l_{v_{α}})_{π^{k}(u),u} = J_{u}^{α} ∘ (l_{h})_{π^{k}(u),u}. If we use the notations N_{α}(u) = (l_{v_{α}})_{π^{k}(u),u}(T_{π^{k}(u)}M) we obtain that (l_{v_{α}})_{π^{k}(u),u} are linear isomorphisms.

Corollary 3.1. The following formulae are true

K_{u}^{(α)} ∘(l_{v_{α}})_{π^{k}(u),u} = id_{T_{π^{k}(u)}M} ∀α ∈ {1, 2, …} and u ∈ E.

Proof. We have (l_{v_{k}})_{π^{k}(u),u} = J_{u}^{k} ∘ (l_{h})_{π^{k}(u),u} = J^{k−α} ∘ J_{u}^{α} ∘ (l_{h})_{π^{k}(u),u} = J_{u}^{k−α} ∘

(l_{v_{α}})_{π^{k}(u),u}. Compound at left to K_{u}^{(k)} in the formula (l_{v_{k}})_{π^{k}(u),u} = J_{u}^{k−α} ∘ (l_{v_{α}})_{π^{k}(u),u}

and using K_{u}^{(k)} ∘(l_{v_{k}})_{π^{k}(u),u} = id_{T_{π^{k}(u)}M} and from (2.1) it results K_{u}^{(α)} ∘(l_{v_{α}})_{π^{k}(u),u} =

id_{T_{π^{k}(u)}M} ∀α ∈ {1, 2, …}.

Acknowledgements. A version of this paper was presented at the First Conference of Balkan Society of Geometers, Politehnica University of Bucharest, September 23-27, 1996.

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“Octav Mayer” Institute of Matematics Iaşi Branch of the Romanian Academy 6600, Iaşi, Romania