Date of Award
Master of Arts in Mathematics
First Reader/Committee Chair
The field of differential geometry is brimming with compelling objects, among which are warped products. These objects hold a prominent place in differential geometry and have been widely studied, as is evident in the literature. Warped products are topologically the same as the Cartesian product of two manifolds, but with distances in one of the factors in skewed. Our goal is to introduce warped product manifolds and to compute their curvature at any point. We follow recent literature and present a previously known result that classifies all flat warped products to find that there are flat examples of warped products which do not arise as the warped product of two flat manifolds. Lastly, using reasonable assumptions we will derive the metric of a spacetime containing one point mass representing the center of a black hole, where the point mass representing the singularity is not modeled. We identify this spacetime as a warped product and use a Weyl curvature invariant to show that the curvature of this spacetime is unbounded near this point mass. This will demonstrate that this model does not allow for such a point to be included in the spacetime, so a black hole really is a “hole” in spacetime. Similarly, but with an opposite conclusion, we also show that the curvature is bounded near the event horizon, suggesting that the event horizon still can be modeled.
Bisson, Angelina, "An Exposition of the Curvature of Warped Product Manifolds" (2023). Electronic Theses, Projects, and Dissertations. 1810.