Generalized Finsler structures on closed 3-manifolds

This is a joint work with Sorin V. Sabau and Gheorghe Pitis. A classical Finsler structure (M;F) is a smooth manifold M endowed with a Banach norm on each tangent space TxM that varies smoothly with the base point all over the manifold, for any x 2 M. A Riemannian manifold is a particular case when each of these Banach norms are induced by a quadratic form. Geometrically, this is equivalent to the choice of a unit sphere in each tangent space, such that one obtains a smooth hypersurface § ‰ TM which has the property that each fiber §x := § \ TxM is a smooth, strictly convex hypersurface in TM which surrounds the origin Ox 2 TxM. Except the preference for local computations, a peculiarity of Finsler structures is that, unlike the Riemannian case, one has no means to specify a canonical Finsler structure on a given manifold, therefore, constructing models for Finslerian struc- tures with given geometrical properties (such as constant flag curvature) is an im- portant topic that rises interesting questions about the local and global generality of such structures. A generalization of classical Finsler structures has been introduced by R. Bryant by defining the notion of (I;J;K)-generalized Finsler structures (see (3)), namely a 3-manifold § endowed with a coframing ! = f!1;!2;!3g satisfying d! 1 = iI! 1 ^ ! 3 + ! 2 ^ ! 3 d! 2 = i! 1 ^ ! 3 d! 3 = K! 1 ^ ! 2 i J! 1 ^ ! 3 : We use here only such structures on 3-manifolds, but these can be defined in any dimension (see (4)). Generalized Finsler structures were introduced with the specific intention of 'micro-localization' of classical Finsler structures that allows separating the local geometrical properties of coframes satisfying certain dierential geometric conditions, or solving PDE's, from the global geometrical properties of the manifolds § or M related with the behavior of the leaf space of certain foliations. There are a lot of questions and problems that this new notion brings about. For instance, the absence of results on the existence of global defined Finsler structures motivates one to study the existence of global defined generalized Finsler struc- tures on a 3-manifold § as well as the case when this is realizable as a classical Finsler structure on a surface M. For the case of constant flag curvature one, the only available constructions are Bryant's. In particular, making use of generalized Finsler structures, he was able to construct for the first time global defined Finsler 1