A note on linear parallel displacements in Finsler geometry

Abstract

The author stated the notion of parallel displacements of vector fields along a curve on the 43-rd Symposium on Finsler geometry at Utsunomiya, 2008. The definition is linear with respect to a vector field. So the author redefines it as "linear" parallel displacement followed in [2](In this book, however, linear parallel displacements are defined by an non-linear connection N).On the other hand, the traditional ways of the definition is not linear with respect to a vector field([1]).Further in the early stages of Finsler geometry, many authors gave its definitions([3])and in the book [4]they were collected. In all cases, however, the studies that the author states in this paper were not investigated. In the first half of section 1, the same contents to [7] and [8] with a little difference are stated and in the second half the necessary and sufficient condition that linear parallel displacements are symmetric for any path is stated. In section 2, the determinant of Φ derived froｍ linear parallel displacements is given and cases that Φ is Identity transformation and an orthogonal one are stated. The terminology and notations are referred to the books [5] and [6]. The author is given very useful suggestions by Prof.T.Aikou and Prof.M.Hashiguchi frequently, and greatly appreciates their kindness.