On osculation of Finsler-type connections

Abstract

The concept of osculating Riemannian spaces of Finsler spaces was introduced by A. Nazim [6], later O. Varga [10] studied it in details and used in the determination of Cartan's connection in Finsler spaces. The method of osculation was extended to the connections of Finsler spaces and those of the line element manifold by A. Mo6r [4], [5] and to areal spaces by L. TamAssy [9]. Roughly speaking, the osculation means that to a rather complicated structure (Finsler structure, Finsler connection), a relatively simple structure (a Riemannian metric, an affine or linear connection) is associated, which approximates the former in some sense. Based on this link there is a hope to obtain results on the more complicated structure. Our aim in this paper is to perform and investigate osculations for Finsler-type connections in a global setting. In Section 1 the necessary preliminaries on connections are given. The Finslertype connections most often used in this paper are Finsler pair connections (see [7], [1]), which are analogous to Matsumoto's Finsler connections [3]. Using the Caftan vector field C, in Section 2 the notion of C-osculating horizontal map is given, and the role of the C1 condition is investigated. In Section 3 the osculation of a connection of the Finsler vector bundle V~ along a section of r is given and its properties are studied. This osculation means the construction of a connection in the base vector bundle 4, whose parallel transport is closely related to the original connection in V4. In the last section the h-osculation, related to the connection is considered, and the link with the classical results [4] is given.