Finsler manifold

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Finsler manifold" – news · newspapers · books · scholar · JSTOR (May 2017) (Learn how and when to remove this message)

In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold M where a (possibly asymmetric) Minkowski norm F(x, −) is provided on each tangent space T_xM, that enables one to define the length of any smooth curve γ : [a, b] → M as

${\displaystyle L(\gamma )=\int _{a}^{b}F\left(\gamma (t),{\dot {\gamma }}(t)\right)\,\mathrm {d} t.}$

Finsler manifolds are more general than Riemannian manifolds since the tangent norms need not be induced by inner products.

Every Finsler manifold becomes an intrinsic quasimetric space when the distance between two points is defined as the infimum length of the curves that join them.

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