Geodesic
Homogeneous variational problems and Lagrangian sections
Communications in Mathematics, Tome 24 (2016) no. 2, pp. 115-123

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MR Zbl
We define a canonical line bundle over the slit tangent bundle of a manifold, and define a Lagrangian section to be a homogeneous section of this line bundle. When a regularity condition is satisfied the Lagrangian section gives rise to local Finsler functions. For each such section we demonstrate how to construct a canonically parametrized family of geodesics, such that the geodesics of the local Finsler functions are reparametrizations.
We define a canonical line bundle over the slit tangent bundle of a manifold, and define a Lagrangian section to be a homogeneous section of this line bundle. When a regularity condition is satisfied the Lagrangian section gives rise to local Finsler functions. For each such section we demonstrate how to construct a canonically parametrized family of geodesics, such that the geodesics of the local Finsler functions are reparametrizations.
Classification : 53C22, 53C60
Keywords: Finsler geometry; line bundle; geodesics 
Saunders, D.J. Homogeneous variational problems and Lagrangian sections. Communications in Mathematics, Tome 24 (2016) no. 2, pp. 115-123. http://geodesic.mathdoc.fr/item/COMIM_2016_24_2_a2/
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