After recalling the structure equations of Finsler structures on surfaces, I define a notion of ‘generalized Finsler structure’ as a way of micro-localizing the problem of describing Finsler structures subject to curvature conditions. I then recall the basic notions of path geometry on a surface and define a notion of ‘generalized path geometry’ analogous to that of ‘generalized Finsler structu...

By the SYZ construction, a mirror pair (X,Xˇ) of complex torus X and partner Xˇ is described as special Lagrangian fibrations X→B Xˇ→B on same base space B. Then, by transform, we can construct simple projectively flat bundle from each affine multisection with unitary local system along it. However, there are ambiguities choices transition functions it, this causes difficulties when try to func...

A scalar valued set function on a Cartesian product of σ-algebras is a Fréchet measure if it is a scalar measure independently in each coordinate. A basic question is considered: is it possible to construct products of Fréchet measures that are analogous to product measures in the classical theory? A Fréchet measure is said to be projectively bounded if it satisfies a Grothendieck type inequali...

In this study we introduce a new tensor in semi-Riemannian manifold, named the M*-projective curvature which generalizes m-projective tensor. We start by deducing some fundamental geometric properties of After that, pseudo symmetric manifolds (PM?S)n. A non-trivial example has been used to show existence such manifold. series interesting conclusions. establish, among other things, that if scala...

In this paper all coordinates in two variables over a Noetherian Q-domain of Krull dimension one are proved to be projectively tame. In order to do this, some results concerning projectively-tameness of polynomials in general are shown. Furthermore, we deduce that all automorphisms in two variables over a Noetherian reduced ring of dimension zero are tame.

Every Finsler metric naturally induces a spray but not so for the converse. The notion sprays of scalar (resp. isotropic) curvature has been known as generalization metrics flag curvature. In this paper, new notion, constant curvature, is introduced and especially it shows that isotropic necessarily even in dimension $$n\ge 3$$ . Further, complete conditions are given constant) to be metrizable...

A Finsler manifold is said to be geodesically reversible if the reversed curve of any geodesic remains a geometrical geodesic. Well-known examples metrics are Randers with closed [Formula: see text]-forms. Another family well-known projectively flat on text]-sphere that have constant positive curvature. In this paper, we prove some and dynamical characterizations metrics, several rigidity resul...

We establish that Hitchin’s connection exist for any rigid holomorphic family of Kähler structures on any compact pre-quantizable symplectic manifold which satisfies certain simple topological constraints. Using Toeplitz operators we prove that Hitchin’s connection induces a unique formal connection on smooth functions on the symplectic manifold. Parallel transport of this formal connection pro...

A Finsler space (M,Σ) is said to be geodesically reversible if each oriented geodesic can be reparametrized as a geodesic with the reverse orientation. A reversible Finsler space is geodesically reversible, but the converse need not be true. In this note, building on recent work of LeBrun and Mason [13], it is shown that a geodesically reversible Finsler metric of constant flag curvature on the...