Finsler Manifold

A Finsler manifold is a differentiable manifold M together with a Finsler function F defined on the tangent bundle of M so that for all tangent vectors v,

• F is smooth on the complement of the zero section of TM.
• F(v) ≥ 0 with equality if and only if v = 0 (positive definiteness).
• F(λv) = λF(v) for all λ ≥ 0 (but not necessarily for λ<0) (homogeneity).
• F(v+w) ≤ F(v)+F(w) for all w at the same tangent space with v (subadditivity).

In other words, F is an asymmetric norm on each tangent space. Typically one replaces the subadditivity with the following strong convexity condition:

• For each tangent vector v, the hessian of F2 at v is positive definite.

Here the hessian of F2 at v is the symmetric bilinear form

also known as the fundamental tensor of F at v. Strong convexity of F2 implies the subadditivity with a strict inequality if u/F(u) ≠ v/F(v). If F2 is strongly convex, then F is a Minkowski norm on each tangent space.

A Finsler metric is reversible if, in addition,

• F(−v) = F(v) for all tangent vectors v.

A reversible Finsler metric defines a norm (in the usual sense) on each tangent space.