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). There are analyzed alternatives to Einstein gravity
(including theories with broken local Lorentz invariance) and shown how general
relativity and modifications can be equivalently re-formulated in Finsler like
variables. We focus on prospects in modern cosmology and Finsler acceleration
of Universe. All known formalisms are outlined - anholonomic frames with
associated nonlinear connection structure, the geometry of the Levi-Civita and
Finsler type connections, all defined by the same metric structure, Einstein
equations in standard form and/or with nonholonomic/ Finsler variables - and
the following topics are discussed: motivation for Finsler gravity; generalized
principles of equivalence and covariance; fundamental geometric/ physical
structures; field equations and nonholonomic constraints; equivalence with
other models of gravity and viability criteria. Einstein-Finsler gravity
theories are elaborated following almost the same principles as in the general
relativity theory but extended to Finsler metrics and connections. Gravity
models with anisotropy can be defined on (co) tangent bundles or on
nonholonomic pseudo-Riemannian manifolds. In the second case, Finsler
geometries can be modelled as exact solutions in Einstein gravity. Finally,
some examples of generic off-diagonal metrics and generalized connections,
defining anisotropic cosmological Einstein-Finsler spaces are analyzed; certain
criteria for Finsler accelerating evolution are analyzed.
