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It is shown that space-time may be not only in a state which is described by Riemann geometry but also in states which are described by Finsler geometry. Transitions between various metric states of space-time have the meaning of phase transitions in its geometric structure. These transitions together with the evolution of each of the possible metric states make up the general picture of space-time manifold dynamics. 95.30.Sf; 98.80.Bp; 98.80.Hw; According to the contemporary cosmological models [1], space-time is Riemannian, i.e. locally isotropic, and preserves its local isotropy in the process of the Universe evolution. At the same time there exist some indirect indications of the fact that nowadays spacetime has a weak relic local anisotropy and therefore it is not unlikely that it is described by Finsler geometry [2] rather than by Riemann geometry. Moreover, if it appears that the Hubble constant has no dipole anisotropy correlated with the dipole anisotropy of the microwave background radiation then it will unequivocally indicate a strong local anisotropy of space-time at the early stage of the Universe evolution. PACSOne of the possible mechanisms of the appearance of a local anisotropy in space-time is the induced phase transition in its geometric structure, caused by the breakdown of higher gauge symmetries and by the appearance of masses in fundamental fields of matter. This involves changes in the metric properties of space-time manifold and it goes over from a state described by Riemann geometry into a state described by Finsler geometry. Since Finslerian space-time differs from Riemannian space-time by the anisotropy of its tangent spaces, in such a transition there occurs a flagrant violation of the local Lorentz

It is shown that the problem of a possible violation of the Lorentz transformations at Lorentz factors γ > 5×10 10 , indicated by the situation which has developed in the physics of ultra-high energy cosmic rays (the absence of the GZK cutoff), has a nontrivial solution. Its essence consists in the discovery of the so-called generalized Lorentz transformations which seem to correctly link the inertial reference frames at any values of γ . Like the usual Lorentz transformations, the generalized ones are linear, possess group properties and lead to the Einstein law of addition of 3-velocities. However, their geometric meaning turns out to be different: they serve as relativistic symmetry transformations of a flat anisotropic Finslerian event space rather than of Minkowski space. Consideration is given to two types of Finsler spaces which generalize locally isotropic Riemannian space-time of relativity theory, e. g. Finsler spaces with a partially and entirely broken local 3D isotropy. The investigation advances arguments for the corresponding generalization of the theory of fundamental interactions and for a specific search for physical effects due to local anisotropy of space-time.24/03/99

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.