# Finsler geometry

Let $\mathcal{M}$ be an $n$-dimensional differential manifold and let $\phi\colon T\mathcal{M}\to\mathbb{R}$ be a function $\phi(x,\xi)$ defined for $x\in\mathcal{M}$ and $\xi\in T_{x}\mathcal{M}$ such that $\phi(x,\cdot)$ is a possibly non symmetric norm on $T_{x}\mathcal{M}$. The couple $(\mathcal{M},\phi)$ is called a Finsler space.

Let us define the $\phi$-length of curves in $\mathcal{M}$. If $\gamma\colon[a,b]\to\mathcal{M}$ is a differentiable curve we define

 $\ell_{\phi}(\gamma):=\int_{a}^{b}\phi(\gamma^{\prime}(t))\,dt.$

So a natural geodesic distance can be defined on $\mathcal{M}$ which turns the Finsler space into a quasi-metric space (if $\mathcal{M}$ is connected):

 $d_{\phi}(x,y):=\inf\{\ell_{\phi}(\gamma)\colon\text{\gamma is a % differentiable curve \gamma\colon[a,b]\to\mathcal{M} such that \gamma(a)=x% and \gamma(b)=y}\}.$

Notice that every Riemann manifold $(\mathcal{M},g)$ is also a Finsler space, the norm $\phi(x,\cdot)$ being the norm induced by the scalar product $g(x)$.

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

Title Finsler geometry FinslerGeometry 2013-03-22 15:03:37 2013-03-22 15:03:37 paolini (1187) paolini (1187) 8 paolini (1187) Definition msc 58B20 msc 53B40 msc 53C60 WulffTheorem