## A study of some differentiable Manifolds

##### Abstract

I undertook the present task with the intention of making some contribution to the 'study of some Differentiable Manifolds'. The Differentiable Manifolds chosen are Finsler, Generalised recurrent Finsler and Special Kawaguchi. In such a type of research, my aim has been to study the various properties of tensors so as to develop the spaces with their help and hence to enrich them.
The whole thesis is divided into five chapters and each chapter is subdivided into articles. The equations are numbered in decimal notation. References for the equations are of the form (c-i.e.) where c, and e stand for the chapter, article and equation number respectively. If coincides with the chapter at hand, it will be omitted. The results which are believed to be new are underlined. The notations used in this work are those of H. Rund (1959) used in his book entitled 'The Differential Geometry of Finsler spaced' and Kawaguchi (1936, 1938).
Chapter 1 is introductory in which the basic concepts of the geometry of Finsler and Special Kawaguchi spaces are given. The results and definitions which are needed in the development of the work are enumerated.
In the first section of Chapter II, a study of the decompositions of projective curvature Tensor field in recurrent Finsler spaces in two different ways has been made whereas the second section has been devoted to define G-2 recurrent projective tensor fields and study the properties of associated recurrence vector and tensor fields in G (2-Fn).
Ricci tensor of hyper surface Fn-1 as a function of a Ricci tensor of the embedding space Fn from the generalised Gauss equation is derived in Chapter III. An expression for the Ricci tensor in the umbilical hyper surface is also obtained along with its particular form.
Chapter IV encompasses the derivation of more general Gauss-Codazzi equations of Classical differential geometry by considering a congruence of curves associated with a hyper surface of a Finsler space whose particular cases coincide with the generalised Gauss-codazzi equations of H. Rund. Gauss-Cidazzi equations for umbilical hyper surface are also studied and their particular cases are also obtained. Last Chapter (v) is devoted to study Special Kawaguchi space. Veblen identities and their relations with Bianchi identities for curvature tensor and Projective curvature tensor are derived. At the end a selected list of references of the books and papers which have been consulted during the completion of this work is given.