Riemannian manifolds and their curvature

Abstract

In this dissertation we study some of the fundamental concepts and results of Riemannian geometry. In particular we look at Riemannian manifolds and their curvature with an emphasis on the Riemannian manifolds that have constant sectional curvature. Starting with the fundamental concepts of Riemannian geometry: the Riemannian metric, the Riemannian connection, geodesics and curvature, this dissertation goes on to cover deep results such as the fundamental theorem of Riemannian geometry, the Hopf-Rinow theorem, the Hadamard theorem and the classification theorem for Riemannian manifolds of constant sectional curvature. Along the way we also cover useful tools such as Jacobi fields. In [3: p. 159], M. P. do Carmo considers a diffeomorphism h : P ! Q between Riemannian manifolds (P; g) and (Q; ~g) which preserves the corresponding (0; 4)-Riemannian curvature tensors R and eR . Referring to R. S. Kulkarni [44] and S. T. Yau [46], he poses a problem of deciding whether h is an isometry. Accordingly, at the end of this dissertation, we look at the problem of deciding if a diffeomorphism between two Riemannian manifolds which preserves the sectional curvature is an isometry