Along with the construction of non-Lorentz-invariant effective field theories, recent studies which are based on geometric models of Finsler space-time become more and more popular. In this respect, the Finslerian approach to the problem of Lorentz symmetry violation is characterized by the fact that the violation of Lorentz symmetry is not accompanied by a violation of relativistic symmetry. That means, in particular, that preservation of relativistic symmetry can be considered as a rigorous criterion of the viability for any non-Lorentz-invariant effective field theory. Although this paper has a review character, it contains (with few exceptions) only those results on Finsler extensions of relativity theory, that were obtained by the authors.

Cosmological singularity and asymptotic behavior of scale factor of generalized cosmological models are analyzed in respect of their structural stability. It is shown, that cosmological singularity is structurally unstable for the majority of models with barotropic perfect fluid with strong energy condition. Inclusion of -term extends the set of structurally stable cosmological models.

No abstract

4-dimensional homogeneous isotropic cosmological models obtained from solutions of vacuum 5dimensional Einstein equations are considered. It is assumed, that the G55-component of the 5-d metric simulates matter in the comoving frame of reference. Observable 4-d metric is defined up to conformal transformations of the metric of 4-d section gµν, with a conformal factor as a function of the component G55. It is demonstrated, that the form of this function determines the matter equation of state. Possible equations of state are analyzed separately for flat, open and close models.

Relations between kinematics, geometry and law of reference frame motion are considered. We show, that kinematical tensors define geometry up to a space functional arbitrariness when integrability condition for spin tensor is satisfied. Some aspects of geometrization principle and geometrical conventionalism of Poincaré are discussed in a light of the obtained results.