Date of Award

5-10-2026

Date Published

June 2026

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Advisor(s)

Jay Hubisz

Second Advisor

William Wylie

Keywords

circle bundle;constant scalar curvature;Killing field;quasi-Einstein manifold;Sasakian Manifold

Subject Categories

Mathematics | Physical Sciences and Mathematics

Abstract

This thesis concerns a generalization of Einstein manifolds called quasi-Einstein manifolds. These manifolds are a triple $(M,g,X)$ where $(M,g)$ is a Riemannian manifold, and $X$ is a smooth vector field on $M$. Quasi-Einstein manifolds are of interest in general relativity, where a special class of quasi-Einstein manifolds are known as near horizon geometries. They are also related to Ricci solitons, which are self-similar solutions of the Ricci flow. In the first set of results in this thesis, we study the existence of Killing fields on compact quasi-Einstein manifolds. In particular, we prove an equivalence between compact quasi-Einstein manifolds of constant scalar curvature, and those where $X$ is Killing. Though the key steps of this proof are outlined in a publication by Ghosh, the author of this thesis was not aware of this work at the time the author proved this result. We further find a necessary and sufficient condition for a compact quasi-Einstein manifold to admit a Killing field, generalizing a result due to Dunajski-Lucietti. In the second set of results in this thesis, we partially classify quasi-Einstein manifolds of constant scalar curvature. In dimension three, we give a complete classification, showing that all compact quasi-Einstein manifolds in dimension three are locally homogeneous. We do this by examining a connection between compact quasi-Einstein metrics of constant scalar curvature and Sasakian manifolds, and then appeal to a classification due to Tanno. From there, we reference the classification of locally homogeneous quasi-Einstein manifolds in dimension three due to Lim. Furthermore, we show that quasi-Einstein metrics of constant scalar curvature, if constructed as a circle bundle over a compact Einstein base, must be odd dimensional, by showing the existence of an almost K\"ahler structure on the base. Finally, we also identify a one parameter family of quasi-Einstein metrics of constant scalar curvature on their circle bundles by considering the canonical variation.

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