%0 Journal Article
%T The Geometry of Tangent Bundles: Canonical Vector Fields
%A Tongzhu Li
%A Demeter Krupka
%J Geometry
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/364301
%X A canonical vector field on the tangent bundle is a vector field defined by an invariant coordinate construction. In this paper, a complete classification of canonical vector fields on tangent bundles, depending on vector fields defined on their bases, is obtained. It is shown that every canonical vector field is a linear combination with constant coefficients of three vector fields: the variational vector field (canonical lift), the Liouville vector field, and the vertical lift of a vector field on the base of the tangent bundle. 1. Introduction Vector fields on tangent bundles belong to basic concepts of pure and applied differential geometry, global analysis, and mathematical physics. Recent research in geometry extends the well-known correspondence of semisprays, sprays, and geodesic sprays to the classical theory of geodesics and connections (see, e.g., [1, 2]). Vector fields on tangent bundles can be considered as an underlying geometric structure for the theory of second-order differential equations [3¨C7]. The semispray theory has been used in the calculus of variations on manifolds to characterize extremal curves of a variational functional as integral curves of the Hamilton or Euler-Lagrange vector fields [2, 4, 8, 9]. Sprays and semisprays also provide a natural framework for extension of classical results of analytical mechanics to contemporary mechanical problems and stimulate a broad research in the global theory of nonconservative systems, symmetries, and the constraint theory (see, e.g., [6, 10¨C12]). This paper is devoted to the structure theory of vector fields on the tangent bundle of a manifold ; our aim will be to classify all canonical vector fields on , independent of any geometric structure or on the topology of . Our main theorem says that every canonical vector field is a linear combination with constant coefficients of three independent vector fields: (a) a variational vector field (the natural lift of a vector field, defined on ), (b) the Liouville vector field, and (c) the vertical lift of a vector field on ; this completes the results obtained in [13]. Another result is the method how the main theorem has been formulated and proved; the concepts we use allow generalizations and applications to analogous problems of discovering and describing canonical geometric objects. Our approach to the problem is based on the theory of jets and differential invariants, and on an observation that the coordinate transformations on naturally define a Lie group , where , the differential group, and its left action on the type fibre of any
%U http://www.hindawi.com/journals/geometry/2013/364301/