Riemannian manifolds and their curvature
Date
2020
Authors
Correia, Artur Muhammad Anize
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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
Description
A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfillment of the requirements for the degree of Master of Science, 2020