Geodesic
Semi-Basic $1$-Forms and Courant Structure for Metrizability Problems
Publications de l'Institut Mathématique, _N_S_98 (2015) no. 112, p. 153

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The metrizability of sprays, particularly symmetric linear connections, is studied in terms of semi-basic 1-forms using the tools developed by Bucataru and Dahl in \cite{b:d}. We introduce a type of metrizability in relationship with the Finsler and projective metrizability. The Lagrangian corresponding to the Finsler metrizability, as well as the Bucataru--Dahl characterization of Finsler and projective metrizability are expressed by means of the Courant structure on the big tangent bundle of $TM$. A byproduct of our computations is that a flat Riemannian metric, or generally an R-flat Finslerian spray, yields two complementary, but not orthogonally, Dirac structures on $T^{\text{big}}TM$. These Dirac structures are also Lagrangian subbundles with respect to the natural almost symplectic structure of $T^{\text{big}}TM$.
DOI : 10.2298/PIM150203020C
Classification : 53C60 53C05;53C15;53C20;53C55;53D18
Keywords: (semi)spray, metrizability, semi-basic 1-form, symplectic form, homogeneity, big tangent bundle, Courant bracket, Dirac structure, isotropic subbundle, Poincaré--Cartan 1-form 
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Mircea Crasmareanu. Semi-Basic $1$-Forms and Courant Structure for Metrizability Problems. Publications de l'Institut Mathématique, _N_S_98 (2015) no. 112, p. 153 . doi: 10.2298/PIM150203020C

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