A function
\({\mathcal {F}}:TM \rightarrow \mathbb {R}_+\) is called the
Finsler (norm) function, if it is a positively 1-homogeneous continuous function, smooth on
\(TM\!\setminus \!\{0\}\) and
$$\begin{aligned} g_{ij}=\frac{1}{2} \frac{\partial ^2 E}{\partial y^i\partial y^j}, \end{aligned}$$
is positive definite at every
\(y\in T_xM \! \setminus \! \{0\}\), where
\(E:=\frac{1}{2}{\mathcal {F}}^2\) is the energy function of
\({\mathcal {F}}\). Formally, the Finsler metric
\(g=g_{ij} \mathrm{d}x^i \otimes \mathrm{d}x^j\) on
M is the same as a Riemannian metric, except that the coefficients
\(g_{ij}\) can depend on the
y variable, that is on the direction too. The hypersurface of
\(T_xM\) defined by
$$\begin{aligned} {\mathcal {I}}_x \! =\! \left\{ \,y \in T_xM : {\mathcal {F}}(y) \! = \! 1\,\right\} . \end{aligned}$$
(1)
is called the
indicatrix at
\(x \in M\). The
geodesics of
\((M, {\mathcal {F}})\) are given by the solutions of a system of second order ordinary differential equations
$$\begin{aligned} \ddot{x}^i = f^i(x,\dot{x}), \end{aligned}$$
in a local coordinate system, where
\(f^i(x,y)\) are determined by the formula
$$\begin{aligned} f^i(x, y) = g^{il}\Bigl (\frac{1}{2}\frac{\partial g_{jk}}{\partial x^l} - \frac{\partial g_{jl}}{\partial x^k} \Bigr ) \, y^jy^k, \qquad i = 1,\ldots , n. \end{aligned}$$
One of the most accessible and demonstrative Finsler metrics comes from Zermelo’s navigation problem where the geodesics corresponds to the paths of shortest travel time in a Riemannian manifold (
M,
h), under the influence of a wind or a current which is represented by a vector field
W. The geometry can be described in terms of a Finslerian setting with the Finsler function
$$\begin{aligned} {\mathcal {F}}(x,y) = \frac{\sqrt{ h_x(W, y)^2 + h_x(y,y)(1-h_x(W,W))} - h_x(W, y)}{1-h_x(W,W)} \end{aligned}$$
(2)
where
\(h_x(W,W)<1\). As the results of [
2] show, there is a one-to-one correspondence between the solutions of the Zermelo navigational problem and Randers metrics.