Differential Geometry - Dynamical Systems

1 We develop the method of anholonomic frames with associated nonlinear connection (in brief, N–connection) structure and show explicitly how geometries with local anisotropy (various type of Finsler–Lagrange–Cartan–Hamilton spaces) can be modelled on the metric–affine spaces. There are formulated the criteria when such generalized Finsler metrics are effectively defined in the Einstein, teleparallel, Riemann–Cartan and metric–affine gravity. We argue that every generic off–diagonal metric (which can not be diagonalized by coordinate transforms) is related to specific N–connection configurations. We elaborate the concept of generalized Finsler–affine geometry for spaces provided with arbitrary N–connection, metric and linear connection structures and characterized by gravitational field strengths, i. e. by nontrivial N–connection curvature, Riemannian curvature, torsion and nonmetricity. We apply an irreducible decomposition techniques (in our case with additional N–connection splitting) and study the dynamics of metric– affine gravity fields generating Finsler like configurations. The classification of basic eleven classes of metric–affine spaces with generic local anisotropy is presented. Pacs: 04.50.+h, 02.40.-k, MSC numbers: 83D05, 83C99, 53B20, 53C60 1 c © S. Vacaru, Generalized Finsler Geometry in Einstein, String and Metric-Affine Gravity, hep–th/ 0310132