Every Finsler manifold becomes an intrinsic quasimetric space when the distance between two points is defined as the infimum length of the curves that join them.
A Finsler manifold is a differentiable manifold M together with a Finsler metric, which is a continuous nonnegative function F: TM → [0, +∞) defined on the tangent bundle so that for each point x of M, In other words, F(x, −) is an asymmetric norm on each tangent space TxM.
The Finsler metric F is also required to be smooth, more precisely: The subadditivity axiom may then be replaced by the following strong convexity condition: 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 F implies the subadditivity with a strict inequality if u⁄F(u) ≠ v⁄F(v).
[1] Let (M, d) be a quasimetric so that M is also a differentiable manifold and d is compatible with the differential structure of M in the following sense: Then one can define a Finsler function F: TM →[0, ∞] by where γ is any curve in M with γ(0) = x and γ'(0) = v. The Finsler function F obtained in this way restricts to an asymmetric (typically non-Minkowski) norm on each tangent space of M. The induced intrinsic metric dL: M × M → [0, ∞] of the original quasimetric can be recovered from and in fact any Finsler function F: TM → [0, ∞) defines an intrinsic quasimetric dL on M by this formula.
Due to the homogeneity of F the length of a differentiable curve γ: [a, b] → M in M is invariant under positively oriented reparametrizations.
The Euler–Lagrange equation for the energy functional E[γ] reads in the local coordinates (x1, ..., xn, v1, ..., vn) of TM as where k = 1, ..., n and gij is the coordinate representation of the fundamental tensor, defined as Assuming the strong convexity of F2(x, v) with respect to v ∈ TxM, the matrix gij(x, v) is invertible and its inverse is denoted by gij(x, v).
Then γ: [a, b] → M is a geodesic of (M, F) if and only if its tangent curve γ': [a, b] → TM∖{0} is an integral curve of the smooth vector field H on TM∖{0} locally defined by where the local spray coefficients Gi are given by The vector field H on TM∖{0} satisfies JH = V and [V, H] = H, where J and V are the canonical endomorphism and the canonical vector field on TM∖{0}.
By Hopf–Rinow theorem there always exist length minimizing curves (at least in small enough neighborhoods) on (M, F).