Finsler Geometry Inspired

Abstract

We start with an arbitrary metric d(x, $\tilde x$) on a differentiable manifold M. (In this note differentiability is always assumed.) For any tangent vector 0 ≠ y ∊ T _x M we choose a curve γ : I → M with $\dot \gamma (0) = y$. Consider the function

$$L(x,y) = \mathop {\lim }\limits_{t \to 0} \frac{{d(x,\gamma (t))}}{t}$$

.