Semi-basic 1-forms and Courant Structure for Metrizability Problems
نویسندگان
چکیده
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 [2]. 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 TbigTM . These Dirac structures are also Lagrangian subbundles with respect to the natural almost symplectic structure of TbigTM .
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