Finsler geometry

Let be an n-dimensional differential manifold and let ϕ:T be a function ϕ(x,ξ) defined for x and ξTx such that ϕ(x,) is a possibly non symmetric norm on Tx. The couple (,ϕ) is called a Finsler space.

Let us define the ϕ-length of curvesPlanetmathPlanetmath in . If γ:[a,b] is a differentiableMathworldPlanetmathPlanetmath curve we define


So a natural geodesic distance can be defined on which turns the Finsler space into a quasi-metric space (if is connectedPlanetmathPlanetmath):

dϕ(x,y):=inf{ϕ(γ):γ is a differentiable curve γ:[a,b] such that γ(a)=x and γ(b)=y}.

Notice that every Riemann manifold (,g) is also a Finsler space, the norm ϕ(x,) being the norm induced by the scalar product g(x).

A finite dimensional Banach space is another simple example of Finsler space, where ϕ(x,ξ):=ξ. Wulff Theorem is one of the most important theorems in this ambient space.

Title Finsler geometryMathworldPlanetmath
Canonical name FinslerGeometry
Date of creation 2013-03-22 15:03:37
Last modified on 2013-03-22 15:03:37
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 8
Author paolini (1187)
Entry type Definition
Classification msc 58B20
Classification msc 53B40
Classification msc 53C60
Related topic WulffTheorem