Miron’s Generalizations of Lagrange and Finsler Geometries: a Self–Consistent Approach to Locally Anisotropic Gravity

Modern gauge theories of high energy physics, investigations in classical and quantum gravity and recent unifications of superstring theories (the so–called M– F– and S– theories) are characterized by a large application of geometric and topological methods. There are elaborated a number of Kaluza–Klein models of space–time and proposed different variants of compactification of higher dimensions. One of the still unsolved important physical problem is the definition of the mechanism of (in general dynamical) such compactification with a rigorous ”selection” of the four dimensional space–time physics from the low energy dynamics of (super) string and supergravitational theories. Another question of challenge of the modern physics is the local anisotropy of background radiation and the development of a consistent scenarious of quantum and classical cosmology. In our works [4, 5, 6, 7] we have concerned the mentioned topics in a more general context of modelling physical processes on (super)vector bundles provided with nonlinear and distinguished connections and metric structures (containing as particular cases both Kaluza–Klein spaces and various extensions of Lagrange and Finsler spaces). We based our investigations on the fundamental results of the famous R. Mirons’s Romanian school of Finsler geometry and its generalizations (as basic references we cite here some monographs and recent works [1]). Perhaps, in the special literature there are cited more than a thousand of works on Finsler geometry, its generalizations and applications. It is well known that a metric more general than a Riemannian one was proposed in 1854 by B. Riemann and studied for the first time, in 1918, by P. Finsler who was a post–graduate student of C. Carathéodory. The purpose of applications of such generalized metrics in thermodynamics and, more generally, in physics was obvious. At present time there are published tens of monographs containing Finsler–like physical theories. Nevertheless the bulk of physicists still persists on a broad implementation of Finsler geometry in modern physics. This skepticism consists not only on a conservatism caused by the predomination of Riemann geometry (with some extensions to Einstein–Cartan–Spaces with torsion and nonmetricity) or by the ”excessive complexity” of Finsler geometry for developing physical theories. At first site the problem of construction of a general