Date of Award

8-2026

Document Type

Thesis

Degree Name

Master of Arts in Mathematics

Department

Mathematics

First Reader/Committee Chair

Corey Dunn

Abstract

Differential geometry is concerned with the properties of calculus and geometry on curved n-dimensional manifolds. As a result, thinking about such a space often runs counter to the Euclidean geometer's intuition of distances, angles, and transformations. This thesis aims to build up to proving an important result in the study of Riemannian manifolds: the Hopf-Rinow theorem.

In Chapter 2, we begin by defining what a manifold is and showing that the collection of directional derivatives at a point on the manifold spans a tangent vector space. After defining a basis and a metric for this space, in Chapter 3, we show that a curve on the manifold whose second derivative is zero, achieved by applying the Levi-Civita connection, meets the criterion for being considered a straight line on the manifold. We define these to be geodesics. By then defining the exponential map in Chapter 4, which identifies straight lines in the tangent space with geodesics on the manifold, and showing this mapping is a local diffeomorphism between the tangent space and the manifold, one can prove that in a neighborhood of a chosen point, geodesics are the shortest path between the chosen point and any other point. We prove other key results, such as if a geodesic can be extended further, then the result is always continuous, and if the exponential map is defined for all of the tangent space and not just a neighborhood of the origin, then geodesics can extend for infinite time.

In Chapter 5, we prove the Hopf-Rinow theorem, which states if a Riemannian manifold is Cauchy complete, then all geodesics extend for infinite time. In Chapter 6, we take a brief dive into psuedo-Riemanninan manifolds: Manifolds in which the Cauchy-Schwartz and Triangle inequalities could possibly run in reverse and we discuss how the Hopf-Rinow theorem can fail as a result. We end by giving a more concrete example of a psuedo-Riemannian manifold for which the Hopf-Rinow theorem fails.

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