Special Finsler Spaces

Abstract

The definition of the Finslerian metric function given in Chapter 1 is of course too general for specific applications. Thus, for example, Riemannian geometry results from a specific choice of the form of the Finslerian metric function. It would be natural to isolate, for the most careful study, those subtypes of Finsler spaces which are nontrivial, yet sufficiently simple for the purposes of particular applications. Such an isolation can be achieved by various methods: by postulating a particular form for a typical Finslerian tensor, a symmetry property of the indicatrix, or the nature of the dependence of the Finslerian metric function on xⁱ or yⁱ etc., not mentioning the choice of any particular metric function. Many special cases have been studied in the literature. In the present chapter, we shall describe a set of special Finsler spaces suitable for the applications studied in this volume, it being necessary for this purpose to develop a rather detailed level of exposition. Most of the formulae that are derived are used in succeeding chapters, while the remainder serve to provide the reader with a helpful guide. All the geometrical definitions and results which will be used in Part C are contained in Sections 2.1–2.4 of the present chapter, the essence of these sections consisting in 19 Propositions proved in all rigour. Three additional Sections 2.5–2.7 present a brief description of some other speical types of Finsler spaces which are attractive in view of their applicability to physical theories.