A brief introduction to Finsler geometry by Dahl M. PDF

By Dahl M.

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Where (xi , ξi ) are local coordinates for T ∗ M \ {0}. If (˜ xi , ξ˜i ) are other standard coordinates for T ∗ M \{0}, then ξi = r i and ξi dxi = ∂∂xx˜ i ξ˜r ∂x d˜ xl = ξ˜i d˜ xi . Hence θ is well defined. 9 (Coordinate independent expression for θ). Let π be the canonical projection π : T ∗ M → M . Then the Poincar´e 1-form θ ∈ Ω1 (T ∗ M ) satisfies θξ (v) = ξ (Dπ)(v) for ξ ∈ T ∗ Q and v ∈ Tξ T ∗ Q . Proof. Let (xi , yi ) be standard coordinates for T ∗ Q near ξ. Then we can ∂ i ∂ i ∂ write ξ = ξi dxi |π(ξ) and v = αi ∂x i |ξ +β ∂y |ξ , Thus (Dπ)(v) = α ∂xi |π(ξ) , i and θξ (v) = yi (ξ)dxi |ξ (v) = ξi αi = ξ (Dπ)(α) .

Ana96] M. Anastasiei, Finsler Connections in Generalized Lagrange Spaces, Balkan Journal of Geometry and Its Applications 1 (1996), no. 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.

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. [MS97] D. McDuff and D. Salamon, Introduction to symplectic topology, Clarendon Press, 1997.

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