Finsler geometry

In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold together with the structure of an intrinsic quasimetric space in which the length of any rectifiable curve γ : [a,b] → M is given by the length functional

where F(x, · ) is a Minkowski norm (or at least an asymmetric norm) on each tangent space T_xM. Finsler manifolds non-trivially generalize Riemannian manifolds in the sense that they are not necessarily infinitesimally Euclidean. This means that the (asymmetric) norm on each tangent space is not necessarily induced by an inner product (metric tensor).

Élie Cartan (1933) named Finsler manifolds after Paul Finsler, who studied this geometry in his dissertation (Finsler 1918).

A Finsler manifold is a differentiable manifold M together with a Finsler function F defined on the tangent bundle of M so that for all tangent vectors v,

In other words, F is an asymmetric norm on each tangent space. Typically one replaces the subadditivity with the following strong convexity condition:

Here the hessian of F² at v is the symmetric bilinear form

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