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 TxM, the second derivative of the function L:TxM->R given by
- 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 . Note that it's reparametrization invariant. Geodesics are curves in M whose length is extremal under functional derivatives.
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