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 $$ \ell_\phi(\gamma) := \int_a^b \phi(\gamma'(t))\, dt. $$

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 $\phi(x,\xi):=\Vert \xi \Vert$ . Wulff Theorem is one of the most important theorems in this ambient space.