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Equivalent definitions for connections in higher order tangent bundles.

1 Introduction

We establish a correspondence one to one between the connections [C.sup.(k-1)] (in the bundle [T.sup.k] M [right arrow] M, used by R. Miron) and [C.sup.(0)] (in the affine bundle [T.sup.k]M 4 [T.sup.k-1] M, used for example in [3]). The coefficients of [C.sup.(0)] are called, following Miron, as dual coefficients, but not related as the coefficients of a connection. In this case we use notations more adequate to our setting, different from [1] or [4]. A sinthetic review of some problems involving higher order connections is performed in [2].

2 Some notions regarding higher order spaces

The higher order space [T.sup.k]M of a manifold M can be viewed as a fibered manifold in k ways, over the bases [T.sup.i] M, 0 [less than or equal to] i [less than or equal to] k - 1. Firstly, let us consider the extreme cases i = 0 and i = k - 1, i.e. the fibered manifolds [T.sup.k] M [[[pi].sub.0] over [right arrow]] M and [T.sup.k] M [[[pi].sub.k-1] over [right arrow]] [T.sup.k-1] M.

The fibered manifold [T.sup.k] M [[[pi].sub.0] over [right arrow]] M was systematically used by R. Miron in the study of non-linear connections, semi-sprays and Lagrangians of order k (for example in [4]).

The affine bundle [T.sup.k] M [[[pi].sub.k-1] over [right arrow]] [T.sup.k-1] M was used in the papers [3] and [8], in the study of affine Hamiltonians of order k, a natural dual counterpart of Lagrangians of order k.

The higher order space [T.sup.k] M is defined according by the change rule of the local coordinates given by [mathematical expression not reproducible], where [mathematical expression not reproducible] is only a local vector field. The Liouville vector field has the local form [mathematical expression not reproducible] and it is a global vector field.

Proposition 2.1. The fibered manifold ([T.sup.k] M,[[pi].sub.k], [T.sup.k-1] M) is an affine bundle, for k [greater than or equal to] 2.

Notice that the coordinates [y.sup.(p)i] are in accordance with [4]-[7].

In the sequel we use the dual k-structures J: T[T.sup.k] M [right arrow] T[T.sup.k] M and J*: T*[T.sup.k] M [right arrow] T*[T.sup.k] M, given by:

[mathematical expression not reproducible], (J)

[mathematical expression not reproducible] (J*)

or

[mathematical expression not reproducible].

It is easy to see that J and J* are (globally defined) endomorphisms on [T.sup.k] M, rank J = rank J* = km.

The vector bundle canonically associated with the affine bundle ([T.sup.k] M, [[pi].sub.k], [T.sup.k-1] M) is the vector bundle [q*.sub.k-1] TM, where [q.sub.k-1]: [T.sup.k-1]M [right arrow] M is [q.sub.k-1] = [[pi].sub.1] [??][[pi].sub.2] [??] ***[??][[pi].sub.k-1]. The fibered manifold ([T.sup.k] M, [q.sub.k], M) is used in [4], [7] in the study of the geometrical objects of order k on M, in particular in the study of Lagrangians of order k on M.

The tensors defined on the fibers of the vertical vector bundle [V.sub.k-1.sup.k] M [right arrow] [T.sup.k] M of the affine bundle ([T.sup.k] M, [[pi].sub.k], [T.sup.k-1] M), or on the fibers of an open fibered submanifold of [V.sub.k-1.sup.k] M [right arrow] [T.sup.k] M, are called d-tensors of order k on M. We denote by [V.sub.j.sup.k] M [right arrow] [T.sup.k] M the vertical bundle of the bundle [T.sup.k] M [right arrow] [T.sup.j] M, where [T.sup.(0)] M = M. Obviously T[T.sup.k] M [contains] [V.sub.0.sup.k] M [contains] [V.sub.1.sup.k] M [contains] *** [contains] [V.sub.k.sup.k-1] M. Their polar bundles are [W.sub.j.sup.k] M = [([V.sub.j.sup.k] M).sup.o], thus [W.sub.0.sup.k] M [subset] [W.sub.1.sup.k] M [subset] * * * [subset] [W.sub.k-1.sup.k]M [subset] T*[T.sup.k] M.

It is easy to see that Im J = [V.sub.0.sup.k], ker J = [V.sub.k-1.sup.k], Im J * = [W.sub.k-1.sup.k], ker J * = [W.sub.0.sup.k].

Notice that the local coordinates ([x.sup.i], [y.sup.(1)i],..., [y.sup.(k)i]) on [T.sup.k] M change according to the rules:

[mathematical expression not reproducible] (2.l)

Local bases of modules of sections in [GAMMA]([V.sub.j.sup.k] M) and [GAMMA]([V.sub.j.sup.k] M) are {[[partial derivative]/[partial derivative][y.sup.(j)i]], [[partial derivative]/[partial derivative][y.sup.(k-1)i]]} and {d[x.sup.i] d[y.sup.(j)i],..., d[y.sup.(j-1)i]} respectively.

3 Connections

A non-linear connection in the affine bundle ([T.sup.k] M, [[pi].sub.k], [T.sup.k-1] M) is a left splitting [C.sup.(k-1)] of the inclusion [V.sub.k-1.sup.k] M [[I.sub.k-1] over [right arrow]] T[T.sup.k] M, i.e. a vector bundle map T[T.sup.k] M [[C.sub.(k-1).sup.k] over [right arrow]] [V.sub.k-1.sup.k] M such that [C.sup.(k-1)] [??] [I.sub.k-1] = [1.sub.[V..sub.k-1.sup.k]] M. Using local coordinates, [C.sup.(k-1)] has the local form:

[mathematical expression not reproducible] (3.1)

A non-linear connection in the bundle ([T.sup.k] M, [[pi].sub.k], M) is a left splitting [C.sup.(0)] of the inclusion [V.sub.0.sup.k] M [[I.sub.0] over [right arrow]] T[T.sup.k] M, i.e. a vector bundle map T[T.sup.k] M [[C.sup.(0)] over [right arrow]] [V.sub.0.sup.k] M such that [C.sup.(0)] [??] [I.sub.0] = [1.sub.[V.sub.0.sup.k]M]. Using local coordinates, [C.sup.(0)] has the local form:

[mathematical expression not reproducible] (3.2)

We are going to establish a one to one correspondence between connections [C.sup.(0)] and [C.sup.(k-1)].

Using constructions performed in [4] (see also [5]), we can perform the following constructions using [C.sup.(0)].

If one denotes by [N.sub.0] [subset] T[T.sup.k] M the horizontal distribution of [C.sup.(0)], then the successive distributions:

[N.sub.1] = J([N.sub.0]),..., [N.sub.k-1] = [J.sup.k-1]([N.sub.0]), [V.sub.k-1.sup.k] M = [J.sup.k]([N.sub.0]) (3.3)

give the decomposition:

T[T.sup.k] M = [N.sub.0] [direct sum] [N.sub.1] [direct sum] *** [direct sum] [N.sub.k-1] [direct sum] [V.sub.k-1.sup.k] (3.4)

and the corresponding projectors denoted by h, [v.sub.1],..., [v.sub.k]. A local base

[mathematical expression not reproducible] for [GAMMA]([N.sub.0]) generate the bases [B.sub.1] = J([B.sub.0]) [subset] [GAMMA]([N.sub.1]),..., [B.sub.k-1] = [J.sup.k-1] (B.sub.0) [subset] [GAMMA]([N.sub.k-1]), B.sub.k = J.sup.k (B.sub.0) [subset] [GAMMA]([V.sub.k-1.sup.k]) and B = [B.sub.0] [union] [B.sub.1] [union] * * * [union] [B.sub.k] is a local base for X([T.sup.k] M), called a Berwald base (see [1]).

Let us consider a dual base of B, B* = {[delta][x.sup.i] = d[x.sup.i], [delta][y.sup.(1)i],..., [delta][y.sup.(k)i]}, where

[delta][y.sup.(k)i] = [M.sub.(k)j.sup.i] d[x.sup.j] + [M.sub.(k-1)j.sup.i] d[y.sup.(1)j] + * * * + [M.sub.(1)j.sup.i] d[y.sup.(k-1)j] + d[y.sup.(k)i]

and [delta][y.sup.(k-1)i] = J*[delta][y.sup.(k)i],..., [delta][y.sup.(1)i] = [(J*).sup.k-1] [delta][y.sup.(k)i], [delta][x.sup.i] = d[x.sup.i] = [(J*).sup.k] [delta][y.sup.(k)i]. The dual coefficients of the connection [C.sup.(0)] are the above coefficients of [delta][y.sup.(k)i]. The duality conditions give a link between the coefficients [N.sub.j.sup.([alpha])i]and [M.sub.([beta])j.sup.i], given by:

[mathematical expression not reproducible] (3.5)

and conversely:

[mathematical expression not reproducible] (306)

The coefficients M change according to the rule:

[mathematical expression not reproducible] (3.7)

The change rule of [delta][y.sup.(k)i] is [delta][y.sup.(k)i'] = [[partial derivative]x.sup.i'/x.sup.i][partial derivative][delta][y.sup.(k)i]. By a straightforward computation one can prove that a connection [C.sup.(k-1)] that has the local form (3.1) can be defined. Its horizontal projector has the form:

[mathematical expression not reproducible].

It can be also proved that conversely, given a connection [C.sup.(k-1)] having the coefficients ([M.sub.([alpha])j.sup.i]), then a (non-linear) connection [C.sup.(0)] with coefficients ([N.sub.j.sup.([alpha])i]) given by relations (3.5) is obtained, having ([M.sub.([alpha])j.sup.i]) as dual coefficients.

A link between the connections [C.sup.(0)] and [C.sup.(k-1)] can be obtained without involving coordinates, as follows. Let us consider a given connection [C.sup.(0)], thus the distribution [N.sub.0] is given. We can construct the distributions (3.3) and the distribution (3.4). We have T[T.sup.k] M = [N.sub.k] [direct sum] [V.sub.k-1.sup.k] M, where [N.sub.k] = [N.sub.0] [direct sum] [N.sub.1] [direct sum] *** [direct sum] [N.sub.k-1]. The polar subbundle [N.sub.k.sup.o] [subset] T*[T.sup.k] M (given by forms in T*[T.sup.k] M that are null on [N.sub.k]) is canonically isomorphic with the dual of the vertical bundle [V.sub.k-1.sup.k] M, thus the inclusion [N.sub.k.sup.o] [subset] T*[T.sup.k] M reverses by duality to a vector bundle epimorphism T[T.sup.k] M [[C.sup.(k-1)] over [right arrow]] [V.sub.k-1.sup.k] M that defines the connection [C.sup.(k-1)].

In order to obtain [C.sup.(0)] from [C.sup.(k-1)] we proceed analogously, following a converse way: The vector bundle epimorphism T[T.sup.k] M [[C.sup.(k-1)] over [right arrow]]M that defines the connection [C.sup.(k-1)] reverses by duality to an inclusion [W.sup.k] = ([V.sub.k-1.sup.k] M)* [subset] T*[T.sup.k] M. We consider the subbundles of T*[T.sup.k] M: [W.sup.k-1] = J*([W.sup.k]),..., [W.sup.1] = [(J*).sup.k-1] ([W.sup.k]), [W.sup.0] = [(J*).sub.k] ([W.sup.k]). We have that T*[T.sup.k] M = [W.sup.0] [direct sum] *** [direct sum] [W.sup.k] and [W.sup.0] = ([[V.sub.0.sup.k] M).sup.o]. Then considering U = [W.sup.1] [direct sum] *** [direct sum] [W.sub.k] we have T*[T.sup.k] M = U [direct sum] ([[V.sub.0.sup.k] M).sup.o] and U is canonically isomorphic with ([V.sub.0.sup.k] M)*. Thus the inclusion U [subset] T*[T.sup.k] M reverses by duality to a vector bundle epimorphism T[T.sup.k] M [[C.sup.(0)] over [right arrow]] [V.sub.0.sup.k] M that defines the connection [C.sup.(0)].

All these prove the following interpretation of the dual coefficients of a non-linear connection of order k, which is the main result of the paper.

Theorem 3.1. There is a one to one correspondence between the connections [C.sup.(0)] in the bundle [T.sup.k] M [[[pi].sub.0] over [right arrow]] M and connections [C.sup.(k-1)] in the bundle [T.sup.k] M [[pi].sub.k-1] over [right arrow] [T.sup.k-1] M.

As we have seen, the correspondence given by the above Proposition associates, in local coordinates, but also in an invariant form, free of coordinates, a connection [C.sup.(k-1)] with a connection [C.sup.(0)] that has the dual coefficients exactly the coefficients of [C.sup.(k-1)].

References

[1] Bucataru I., Linear connections for systems of higher order differential equations, Houston Journal of Mathematics, 31 (2005), 315-332.

[2] Bucataru I., A setting for higher order differential equation fields and higher order lagrange and finsler spaces, Journal of Geometric Mechanics, 5 (2013), 257-279.

[3] Crampin M., Sarlet W., Cantrijn F., Higher Order differential equations and higher order lagrangian mechanics, Math. Proc. Camb. Phil. Soc., 86 (1986), 565-587.

[4] Miron R., The Geometry of Higher Order Lagrange Spaces. Applications to Mechanics and Physics, Kluwer, Dordrecht, FTPH, 1997.

[5] Miron R., The Geometry of Higher-Order Hamilton Spaces. Applications to Hamiltonian Mechanics. Kluwer, Dordrecht, FTPH, 2003.

[6] Miron R., Anastasiei M., The Geometry of Lagrange Spaces: Theory and Applications, Kluwer Acad. Publ., 1994.

[7] Miron R., Atanasiu Gh., Differential geometry of the k-osculator bundle, Rev.Roum.Math.Pures Appl., 41 (1996), 205-236.

[8] Popescu P., Popescu M., Affine Hamiltonians in Higher Order Geometry, Int. J. Theor. Phys., 46 (2007), 2531-2549.

Marcela Popescu

Department of Applied Mathematics, University of Craiova, 13 A.I.Cuza st. 200585 Craiova, Romania

E-mail: marcelacpopescu@yahoo.com

Paul Popescu

Department of Applied Mathematics, University of Craiova, 13 A.I.Cuza st. 200585 Craiova, Romania

E-mail: paul_p_popescu@yahoo.com
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Author:Popescu, Marcela; Popescu, Paul
Publication:Libertas Mathematica
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Date:Jun 1, 2017
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