  Finslerian extension of the theory of relativity implies that space-time can
be not only in an amorphous state which is described by Riemann geometry but
also in ordered, i.e. crystalline 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. It is shown that there are
only two types of curved Finslerian spaces endowed with local relativistic
symmetry. However the metric of only one of them satisfies the correspondence
principle with Riemannian metric of the general theory of relativity and
therefore underlies viable Finslerian extension of the GR. Since the existing
purely geometric approaches to a Finslerian generalization of Einstein's
equations do not allow one to obtain such generalized equations which would
provide a local relativistic symmetry of their solutions, special attention is
paid to the property of the specific invariance of viable Finslerian metric
under local conformal transformations of those fields on which it explicitly
depends. It is this property that makes it possible to use the well-known
methods of conventional field theory and thereby to circumvent the
above-mentioned difficulties arising within the framework of purely geometric
approaches to a Finslerian generalization of Einstein's equations.
