Semi-basic 1-forms and courant structure for metrizability problems

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 at Riemannian metric, or generally an R-at 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.