The length and the relative length of tangent vectors of finsler spaces

Abstract

We consider the length of a vector in a Finsler space with the fundamental function L( x, y). The length of a vector X is usually defined as the value L( x, X) of L. On the other hand, we have an essential tensor g ij ( x, y), called the fundamental tensor, and the concept of relative length | X y | of X may be introduced by | X| y y = g ij ( x, y) X i X j with re spect to a supporting element y. The question arises whether is L( x, X) the minimum of | X| y or not? If there exists a supporting element y satisfying | X| y < L( x, X), then a curve x( t) in the Finsler space will be measured shorter than the usual length, by integrating | dx/ dt| y with the field of such supporting element y( t) along the curve.