By Dahl M.

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Proof. Suppose E is stationary for a curve c. 6 implies that cˆ is a integral curve of G. 4. 8. 17. If γ : I → T M \ {0} is an integral curve of G/F , then π ◦ γ is a stationary curve for E. Conversely, if c is a stationary curve for E, then λ = F ◦ cˆ is constant and c ◦ M1/λ (see below) is an integral curve of G/F . If s > 0, we denote by Ms the mapping Ms : t → st, t ∈ R. 35 Proof. Let c : I → T M \ {0} be an integral curve of G/F . If c = (x, y), then dxi dt dy i dt = yi , λ = −2 Gi ◦ c , λ where λ = F ◦γ > 0 is constant.

14. In a Finsler space (M, F ), the Hilbert 1-form η and the geodesic spray G satisfy 1 dη(G, ·) = d( F 2 ), 2 so X 1 F 2 = G. What is more, XF = G/F . 2 Proof. Using dη = − ∂gij i r y dx ∧ dxj + gij dxi ∧ dy j , ∂xr we obtain dη(G, ·) = = ∂gij ∂gis 1 ∂F 2 i i j i s − dy y y + 2g G dx + is ∂xs ∂xj 2 ∂y i 1 ∂F 2 i 1 ∂F 2 i dx + dy . 2 ∂xi 2 ∂y i The second claim follows since ιXF ω = dF = ιG/F ω. 14 state that dη is preserved under the flow of G. 15. F is a constant on integral curves of G and G/F .

Equation (47) follows from equation (48). The next proposition shows how stationary curves of E are transformed under the Legendre transformation. 19. 18, and π is the canonical projection π : T ∗ M → M . 1. If γ : I → T ∗ M \ {0} is an integral curve of Xh , then π ◦ γ is a stationary curve for E. 36 2. If c : I → M is a stationary curve for E, then λ = F ◦ cˆ is constant and L −1 ◦ c ◦ M1/λ is an integral curve of Xh . Proof. In the first claim, L ◦ γ is an integral curve of X F . 17. The other proof is similar.

### A brief introduction to Finsler geometry by Dahl M.

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