On the Parallel Displacement and Parallel Vector Fields in Finsler Geometry

In Finsler geometry, the notion of parallel for Finsler tensor fields is defined, already. In this paper, however, for vector fields on a base manifold, the author studies the notion of parallel. In the last section, the obtained results are shown as compare corresponding properties to them of Riemannian geometry. Introduction The author gave the definition (1.1) of the parallel displacement along a curve in [7] and explained the geometrical meaning by HTM(the horizontal subbundle of TTM) in it. Further, the author proved that the inverse vector field was parallel along the inverse curve under some conditions, in addition, that the inner product of two parallel vector fields was preserved along a path. Next,the notion of the autoparallel curves was stated and, last, the author formulated the notion of the geodesics by using the Cartan Finsler connection and the notions of the path and autoparallel curves. In this paper, the author studies the notion of parallel displacement for vector fields on a base manifold. The notion of parallel for Finsler tensor fields is already defined([3],[5],[6]) as same as done the notion of parallel displacement along curves([2],[1]). However, the study in detail of parallel vector field on the base manifold are not done, the author thinks so. As the first step of the study in detail, the author investigates the geometrical meaning of parallel in HTM (Proposition 3.1) and from the obtained results the author have the conditions for Finsler connections (Theorem 3.1). Lastly, in Riemannian geometry, parallel 2000 Mathematics Subject Classification. 53B40.