Date of Award

6-2019

Document Type

Thesis

Degree Name

Master of Arts in Mathematics

Department

Mathematics

First Reader/Committee Chair

Dunn, Corey

Abstract

A manifold is a Hausdorff topological space that is locally Euclidean. We will define the difference between a Riemannian manifold and a pseudo-Riemannian manifold. We will explore how geodesics behave on pseudo-Riemannian manifolds and what it means for manifolds to be geodesically complete. The Hopf-Rinow theorem states that,“Riemannian manifolds are geodesically complete if and only if it is complete as a metric space,” [Lee97] however, in pseudo-Riemannian geometry, there is no analogous theorem since in general a pseudo-Riemannian metric does not induce a metric space structure on the manifold. Our main focus will be on a family of manifolds referred to as a generalized plane wave manifolds. We will prove that all generalized plane wave manifolds are geodesically complete.

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