Finsler Manifold - Geodesics

Geodesics

Due to the homogeneity of F the length

of a differentiable curve γ:→M in M is invariant under positively oriented reparametrizations. A constant speed curve γ is a geodesic of a Finsler manifold if its short enough segments γ| are length-minimizing in M from γ(c) to γ(d). Equivalently, γ is a geodesic if it is stationary for the energy functional

in the sense that its functional derivative vanishes among differentiable curves γ:→M with fixed endpoints γ(a)=x and γ(b)=y.