We propose a new framework for constructing geometric and physical models on nonholonomic manifolds provided both with Clifford – Lie algebroid symmetry and nonlinear connection structure. Explicit parametrizations of generic off–diagonal metrics and linear and nonlinear connections define different types of Finsler, Lagrange and/or Riemann–Cartan spaces. A generalization to spinor fields and D...

A linear connection on a Finsler manifold is called compatible to the function if its parallel transports preserve Finslerian length of tangent vectors. Generalized Berwald manifolds are equipped with connection. In paper we present general and intrinsic method characterize connections dimension three. We prove that not unique then indicatrices must be Euclidean surfaces revolution. The surplus...

The starting point of the famous structure theorems on Berwald spaces due to Z.I. Szabó [4] is an observation on the Riemann-metrizability of positive definite Berwald manifolds. It states that there always exists a Riemannian metric on the underlying manifold such that its Levi-Civita connection is just the canonical connection of the Berwald manifold. In this paper we present a new elementary...

Let x : (M, F ) →֒ (V n+1, F ) be a simply connected hypersurface in a Minkowski space (V n+1, F ). In this paper, using the Gauss formula of Chern connection on Finsler submanifolds, we shall prove that if x(p) is normal to Tp(M)(∀p ∈ M), then M with the induced metric is isometric to the standard Euclidean sphere.

This article uses the Berwald connection exclusively, together with its two curvatures, to cut an efficient path across the landscape of Finsler geometry. Its goal is to initiate differential geometers into two key research areas in the field: the search for unblemished “unicorns” and the study of Ricci flow. The exposition is almost self-contained.

Finsler geometry is a natural and fundamental generalization of Riemann geometry. The Finsler structure depends on both coordinates and velocities. It is defined as a function on tangent bundle of a manifold. We use the Bianchi identities satisfied by Chern curvature to set up a gravitation theory in Berwald-Finsler space. The geometric part of the gravitational field equation is nonsymmetric i...

Kinematic and geometrical aspects of the connection between energy and inertial mass are considered. Transformations of coordinate and time are obtained in a two-dimensional flat space. For this case the Euclidean, pseudo-Euclidean and Galilean kinematics are considered. A new interval in a flat fourdimensional anisotropic Finsler space is found. Under certain assumptions the known results foll...

In this paper, by the Gauss equation of the induced Chern connection for Finsler submanifolds, we prove that if M is an umbilical hypersurface of a Minkowski space (V , F ), then either M is a Riemannian space form or a locally Minkowski space. AMS subject classifications: 53C60, 53C40

Based on the analogue spacetime programme, and many other ideas currently mooted in “quantum gravity”, there is considerable ongoing speculation that the usual pseudoRiemannian (Lorentzian) manifolds of general relativity might eventually be modified at short distances. Two specific modifications that are often advocated are the adoption of Finsler geometries (or more specifically, pseudo-Finsl...

Modern gauge theories of high energy physics, investigations in classical and quantum gravity and recent unifications of superstring theories (the so–called M– F– and S– theories) are characterized by a large application of geometric and topological methods. There are elaborated a number of Kaluza–Klein models of space–time and proposed different variants of compactification of higher dimension...