Finsler manifold

In geometry, Finsler manifolds are a generalization of Riemannian manifolds.

They are named after Paul Finsler.

Definition

A Finsler manifold is a differentiable manifold with a smooth outside the zero -section function such that for all:

• With equality only for
• For all
• .

The Finsler manifold is called symmetric if for all.

Examples

• Normed vector spaces, if the norm outside of the zero vector is smooth.
• Riemannian manifolds: set.
• Convex sets with the Hilbert metric: set for.

Length and volume

The length of a curve is defined by rectifiable

The form of a volume -dimensional Finsler manifold is defined as follows. Be, be a basis of the dual basis. Be the Euclidean volume. The volume shape is then given by

Where the Euclidean volume of the unit ball in the designated. The Busemann volume of a measurable amount is defined by.