Finsler manifold

From Exampleproblems

In mathematics, a Finsler manifold is a differential manifold M with a Banach norm defined over each tangent space such that the Banach norm as a function of position is smooth, usually it is assumed to satisfy the following regularity condition:

For each point x of M, and for every nonzero vector v in the tangent space T_xM, the second derivative of the function L:T_xM → R given by

$L(w)=\frac{1}{2}\|w\|^2$

at v is positive definite.

Riemannian manifolds (but not pseudo-Riemannian manifolds) are special cases of Finsler manifolds.

The length of γ, a differentiable curve in M, is given by

$\int \left\|\frac{d\gamma}{dt}(t)\right\| dt.$

Length is invariant under reparametrization. With the above regularity condition, geodesics are locally length-minimizing curves with constant speed, or equivalently curves in whose energy function

$\int \left\|\frac{d\gamma}{dt}(t)\right\|^2 dt.$