Finsler geometry

Let $ℳ$ be an$n$-dimensional differential manifold and let $\varphi :Tℳ\to ℝ$ be a function $\varphi (x,\xi )$ defined for $x\in ℳ$ and $\xi \in T_xℳ$ such that $\varphi (x,\cdot )$ is a possibly non symmetric norm on $T_xℳ$. The couple $(ℳ,\varphi )$ is called a Finsler space.

Let us define the $\varphi$-length of curves in $ℳ$. If $\gamma :[a,b]\to ℳ$ is a differentiable curve we define

$${\mathrm{\ell }}_{\varphi }(\gamma ):={\int }_a^b\varphi ({\gamma }^{\prime }(t))𝑑t.$$

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

$$d_{\varphi }(x,y):=inf\{{\mathrm{\ell }}_{\varphi }(\gamma ):\gamma \text{ is a differentiable curve }\gamma :[a,b]\to ℳ\text{ such that }\gamma (a)=x\text{ and }\gamma (b)=y\}.$$

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

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