Semi-basic $1$-forms and Courant structure for metrizability problems

Mircea Crasmareanu

Abstract: 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 $T^{\text{big}}TM$.

Keywords: (semi)spray; metrizability; semi-basic 1-form; symplectic form; homogeneity; big tangent bundle; Courant bracket; Dirac structure; isotropic subbundle; Poincaré–Cartan 1-form

Classification (MSC2000): 53C60; 53C05;53C15;53C20;53C55;53D18

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