New PDF release: A brief introduction to Finsler geometry

By Dahl M.

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1, 1–10. [Con93] L. Conlon, Differentiable manifolds: A first course, Birkh¨auser, 1993. N. Dzhafarov and H. Colonius, Multidimensional fechnerian scaling: Basics, Journal of Mathematical Psychology 45 (2001), no. 5, 670–719. S. Ingarden, On physical applications of finsler geometry, Contemporary Mathematics 196 (1996). [Kap01] E. Kappos, Natural metrics on tangent bundle, Master’s thesis, Lund University, 2001. [KT03] L. Kozma and L. Tam´assy, Finsler geometry without line elements faced to applications, Reports on Mathematical Physics 51 (2003).

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. The first equation implies that c = λπ ◦ γ. Since Gi is 2-homogeneous, it follows that d2 xi + 2Gi (π ◦ c) = 0, dt2 so π ◦ c is a stationary curve for E.

6. Suppose (M, ω) is a symplectic manifold, and X H is a Hamiltonian vector field corresponding to a function H : M → R. Furthermore, suppose Φ : I × U → M is the local flow of X H defined in some open 31 U ⊂ M and open interval I containing 0. Then for all x ∈ U , t ∈ I, we have (Φ∗t ω)x = ωx , where Φt = Φ(t, ·). Proof. As Φ0 = idM , we know that the relation holds, when t = 0. Therefore, let us fix x ∈ U , a, b ∈ Tx M , and consider the function r(t) = (Φ∗t ω)x (a, b) with t ∈ I. Then r (t) = = = d r(s + t) ds s=0 d ∗ (Φs ω)y (a , b ) ds LXH ω y (a , b ), s=0 where y = Φt (x), a = (DΦt )(a), b = (DΦt )(b), and the last line is the definition of the Lie derivative.

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A brief introduction to Finsler geometry by Dahl M.


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