Differential geometry of projectively flat Finsler spaces : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Palmerston North, New Zealand
The aim of this thesis is to study the theory of Finsler spaces by considering the following main objectives. (i) To present the basic concepts of Finsler geometry including connections, flag curvature, projective changes, Randers spaces and Finsler spaces with other types of (α,β)-metric, where α is a Riemannian metric and β is a one-form. (ii) To introduce a Riemannian space of non-zero constant sectional curvature by considering a locally projectively flat Finsler space. The requirement for the Riemannian connection to be metric compatible gives a system of partial differential equations. Further, we compute two standard Riemannian metrics of non-zero constant sectional curvature by choosing two solutions of this system of partial differential equations. (iii) To give two examples of locally projectively flat Randers metrics of scalar curvature by using a Riemannian metric computed in (ii) to illustrate the fact that some locally projectively flat Randers metrics of scalar curvature do not have isotropic S-curvature. We also prove that the scalar curvature of a Randers metric is not necessarily a constant if the metric has isotropic S-curvature and closed one-form by using an example. (iv) To find necessary and sufficient conditions for Finsler spaces with various types of (α,β)-metric to be locally projectively flat and determine whether the conditions, a Riemannian metric (α) is locally projectively flat and a one-form (β) is closed, can occur at the same time in the locally projectively flat Finsler spaces with various types of (α,β)-metric.