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Let F be a Finsler norm on M . Then the Hilbert 1-form η ∈ Ω1 (T M \ {0}) is defined as η|y = −gij (y)y i dxj |y , y ∈ T M \ {0}. The next proposition shows that η is globally defined. 13. If L : T M → T ∗ M is the Legendre transformation induced by a Finsler norm, then L ∗ θ = η, dη is a symplectic form for T M \ {0}, and L : T M \ {0} → T ∗ M \ {0} is a symplectic mapping. Proof. The first claim follows directly from the definitions by expanding the left hand side. Since dθ is non-degenerate, it follows that dη is nondegenerate.

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. Using Cartan’s formula, L X = ιX ◦ d + d ◦ ιX , we have LXH ω = ιXH dω + dιXH ω = ιXH 0 + ddH = 0, so r (t) = 0, and r(t) = r(0) = ωx (a, b). Suppose M, N are manifolds, Ψ : M → N is a diffeomorphism. Then the pullback of Ψ for vector fields is the mapping Ψ∗ : X (N ) → X (M ) , Y → (DΨ−1 ) ◦ Y ◦ Ψ. 7. Suppose (M, ω), (N, η) are symplectic manifolds, Φ : M → N is a symplectic mapping such that Φ ∗ η = ω, and h : N → R is a smooth function.

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). [MA94] R. Miron and M. Anastasiei, The geometry of lagrange spaecs: Theory and applications, Kluwer Academic Press, 1994.

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A brief introduction to Finsler geometry by Dahl M.


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