Finsler Metric

A continuous real function $L(x,y)$ defined on the Tangent Bundle $T(M)$ of an $n$-D Differentiable Manifold $M$ is said to be a Finsler metric if

1. $L(x,y)$ is Differentiable at $x\not=y$,

2. $L(x,\lambda y)=\vert\lambda\vert L(x,y)$ for any element $(x,y)\in T(M)$ and any Real Number $\lambda$,

3. Denoting the Metric

$$g_{ij}(x,y)={1\over 2}{\partial^2[L(x,y)]^2\over\partial y^i\partial y^j},$$

then $g_{ij}$ is a Positive Definite Matrix.

A Differentiable Manifold $M$ with a Finsler metric is called a Finsler Space.

See also Differentiable Manifold, Finsler Space, Tangent Bundle

References

Iyanaga, S. and Kawada, Y. (Eds.). ``Finsler Spaces.'' §161 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 540-542, 1980.