Geodesics: Covariant Differentiation

Abstract

Having described the more essential metric properties of the local tangent spaces, we now proceed to study the underlying manifold X _n , which we shall henceforth denote by F _n in order to stress the fact that a Finsler metric has been imposed upon it. Some of the most fundamental properties of F _n are described by the extremals of the problem in the calculus of variations which provides us with our metric. Thus we shall first derive the differential equations satisfied by the extremals — or ‘‘geodesies” — of F _n ; this will be done not by means of the customary method involving the first variation of the length integral, but by means of a method specially adapted to illustrate clearly the geometrical background underlying this derivation. The basic problem in a geometry such as that of F _n is the investigation of the mutual relationship between tangent spaces attached to neighbouring points of F _n : more precisely, one seeks to establish geometrically meaningful mappings of one such tangent space onto another. In more elementary terms this problem may be formulated by posing the question as to the type of conditions which must be satisfied such that two vectors belonging to distinct but “neighbouring” tangent spaces may be described as being parallel. This question is by no means trivial: analytically it presents itself through the phenomenon that the ordinary derivative of a tensor is not in general also a tensor. It is from this point of view that we shall carry out a preliminary analysis of this problem in the present chapter.