Principles of Einstein–finsler Gravity and Perspectives in Modern Cosmology

We study the geometric and physical foundations of Finsler gravity theories with metric compatible connections defined on tangent bundles, or (pseudo) Riemannian manifolds, endowed with nonholonomic frame structure. Several generalizations and alternatives to Einstein gravity are considered, including modifications with broken local Lorentz invariance. It is also shown how such theories (and general relativity) can be equivalently re-formulated in Finsler like variables. We focus on prospects in modern cosmology and Finsler acceleration of Universe. Einstein–Finsler gravity theories are elaborated following almost the same principles as in the general relativity theory but extended to Finsler metrics and connections. Finally, some examples of generic off-diagonal metrics and generalized connections, defining anisotropic cosmological Einstein–Finsler spaces are analyzed; certain criteria for the Finsler accelerating evolution are formulated.