Finsler manifold

Remarks

Example: if $g$ is a Riemannian metric on $M$, then $F(v) = \sqrt{g(v, v)}$ is a Finsler function. It is smooth away from $0$, homogeneous of degree $1$, and the Hessian of $\tfrac{1}{2}F^2 = \tfrac{1}{2} g(v, v)$ on $T_pM$ is $g_p$ itself, which is positive definite.

Non-example: on $\R^2$, $F(x, v) = \abs{v_1} + \abs{v_2}$ is positively homogeneous, but it is not smooth along the lines $v_1 = 0$ and $v_2 = 0$, so clause 1 fails.

Clauses 2 and 3 imply that $F(v) > 0$ for $v \ne 0$ and that $F$ satisfies the triangle inequality on each tangent space. The function $F$ need not be symmetric: $F(-v) = F(v)$ is not required, and Randers metrics are standard examples with $F(-v) \ne F(v)$. The length of a piecewise smooth curve $\gamma$ is $\int F(\gamma'(t)) \, dt$.