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The aim of this book is to present the basic concepts of Finsler geometry including connections, flag curvature, projective changes, Randers spaces and other types of Finsler spaces. In this book, we introduce a Riemannian space of non-zero constant sectional curvature by considering a locally projectively flat Finsler space and compute two standard Riemannian metrics of non-zero constant sectional curvature by choosing two solutions of a system of partial differential equations. This book also presents two examples of locally projectively flat Randers metrics of scalar curvature by using the Riemannian metric to illustrate the fact that some locally projectively flat Randers metrics of scalar curvature do not have isotropic S-curvature. Finally we give necessary and sufficient conditions for Finsler spaces with various types of (alpha,beta)-metric to be locally projectively flat and determine whether the conditions, a Riemannian metric (alpha) is locally projectively flat and a one-form (beta) is closed, can occur at the same time in the locally projectively flat Finsler spaces with various types of (alpha,beta)-metric.
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