We outline an unified approach to geometrization of Lagrange mechanics, Finsler geometry and geometric methods of constructing exact solutions with generic off-diagonal terms and nonholonomic variables in gravity theories. Such geometries with induced almost symplectic structure are modelled on nonholonomic manifolds provided with nonintegrable distributions defining nonlinear connections. We introduce the concept of Lagrange-Fedosov spaces and Fedosov nonholonomic manifolds provided with almost symplectic connection adapted to the nonlinear connection structure. We investigate the main properties of generalized Fedosov nonholonomic manifolds and analyze exact solutions defining almost symplectic Einstein spaces.

This is a survey on bi-Lagrangian manifolds, which are symplectic manifolds endowed with two transversal Lagrangian foliations. We also study the non-integrable case (i.e., a symplectic manifold endowed with two transversal Lagrangian distributions). We show that many different geometric structures can be attached to these manifolds and we carefully analyse the associated connections. Moreover, we introduce the problem of the intersection of two leaves, one of each foliation, through a point and show a lot of significative examples.Comment: 30 page

We study several linear connections (the first canonical, the Chern, the well adapted, the Levi Civita, the Kobayashi-Nomizu, the Yano, the Bismut and those with totally skew-symmetric torsion) which can be defined on the four geometric types of $(J^2=\pm1)$-metric manifolds. We characterize when such a connection is adapted to the structure, and obtain a lot of results about coincidence among connections. We prove that the first canonical and the well adapted connections define a one-parameter family of adapted connections, named canonical connections, thus extending to almost Norden and almost product Riemannian manifolds the families introduced in almost Hermitian and almost para-Hermitian manifolds. We also prove that every connection studied in this paper is a canonical connection, when it exists and it is an adapted connection.Comment: Corrected typos and updated reference

In this paper, we study the well adapted connection attached to a (J 2 = ±1)-metric manifold, proving it exists for any of the four geometries and obtaining a explicit formula as a derivation law. Besides we characterize the coincidence of the well adapted connection with the Levi Civita and the Chern connections.

An almost Golden Riemannian structure (ϕ, g) on a manifold is given by a tensor field ϕ of type (1,1) satisfying the Golden section relation ϕ 2 = ϕ + 1, and a pure Riemannian metric g, i.e., a metric satisfying g(ϕX, Y ) = g(X, ϕY ). We study connections adapted to such a structure, finding two of them, the first canonical and the well adapted, which measure the integrability of ϕ and the integrability of the G-structure corresponding to (ϕ, g).