Finsler geometry, relativity and gauge theories. (English) Zbl 0576.53001

Asanov, G. S.

Fundamental Theories of Physics, 12. Dordrecht-Boston-Lancaster-Tokyo: D. Reidel Publishing Company, a member of the Kluwer Academic Publishers Group. X, 370 p. Dfl. 195.00; $ 64.00; £54.25 (1985).

The purpose of the book is to elaborate a systematic study of the main techniques for extending general relativity based on the geometry of Finsler spaces. The first two chapters stand for a rapid introduction to Finsler geometry. Chapter 3 is concerned with the general theory of invariance identities and its applications to constructions of connection coefficients and fundamental tensor densities. In chapters 4 and 5 two Finslerian approaches are formulated to obtain generalizations of the physical field equations. The first approach is based on the concept of osculation, according to which a vector field \(y^ i(x)\) is going to be substituted for the directional variable \(y^ i\) any time when a Lagrangian density is constructed. The second approach is based on the notion of indicatrix in a Finsler space, which is in fact a hypersurface in the tangent space to a Finsler space, defined by the equation \(F(x,y)=1\), where F is the Finslerian metric function. Some problems of classical mechanics are studied in chapter 6 from the Finslerian viewpoint. Chapter 7 is dealing with Finslerian extension of the special principle of relativity and Finslerian kinematics.
There are added two appendices. In the first one the author describes the Rund theory with respect to the concept of direction-dependent connection. The second one is concerned with a generalization of both, the Finslerian gauge ideas from chapter 5 and from the first appendix. Each chapter ends with some notes and problems. Finally, the author provides to the reader an up to day list of publications on Finsler geometry. We conclude by saying that Asanov’s book is an important contribution to literature and should benefit both experts and novices in applications of Finsler geometry.

Reviewer: A.Bejancu

MSC:

53-02	Research exposition (monographs, survey articles) pertaining to differential geometry
53B40	Local differential geometry of Finsler spaces and generalizations (areal metrics)
53B50	Applications of local differential geometry to the sciences
83D05	Relativistic gravitational theories other than Einstein’s, including asymmetric field theories

Keywords:

osculation concept; indicatrix; general relativity; Finsler spaces; direction-dependent connection; Finsler geometry