Finsler manifold

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 and satisfies the following property:

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

<math>L(\bold{w})=\frac{1}{2}\|w\|^2<math>

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

<math>\int \left\|\frac{d\gamma}{dt}(t)\right\| dt<math>.

Note that this is reparametrization-invariant. Geodesics are curves in M whose length is extremal under functional derivatives.