Date of Award

5-10-2026

Date Published

June 2026

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Advisor(s)

William Wylie

Keywords

Differential geometry;Homogeneous spaces;Lie groups;Manifolds;Quasi-Einstein metric;Riemannian Geometry

Subject Categories

Mathematics | Physical Sciences and Mathematics

Abstract

This thesis investigates a classic question in Riemannian geometry: What is the best metric to put on a Riemannian manifold? It combines two projects that explore different aspects of this question. The first project studies nilpotent and unimodular solvable Lie groups that admit m-quasi-Einstein metrics (M,g,X) with X a left-invariant vector field, which we call totally left-invariant quasi-Einstein metrics. We give a complete classification of nilpotent Lie groups admitting such metrics, showing that this occurs if and only if the group is isomorphic to Heisenberg Lie group. For unimodular solvable Lie groups S, we prove that the existence of a non-flat totally left-invariant quasi-Einstein metric forces the center of S to be one-dimensional. Furthermore, under the additional assumption that the adjoint operator of the Lie algebra of S is a normal derivation, we obtain a full classification: S is isomorphic to semi-direct product of Euclidean space and Heisenberg Lie group. As an application, we show that the only near-horizon geometries on a compact nilmanifold are of the form $\Gamma \backslash H_n$, where $H_n$ is the n-dimensional Heisenberg Lie group. The second project completes the structure theory of homogeneous conformally Einstein manifolds by resolving the final open case left by Petersen--Wylie: one-dimensional extensions of homogeneous spaces that admit a conformally Einstein metric. On such spaces, the conformally Einstein metric is in fact a gradient m-quasi-Einstein metric with m = 2-n. We prove that if a one-dimensional extension of a homogeneous space admits a gradient m-quasi-Einstein metric, then the base is a Ricci soliton when $\lambda\neq 0$, and flat when $\lambda=0$. This establishes that every simply-connected homogeneous non-trivial conformally Einstein manifold is isometric to either a constant curvature space, a product of Einstein space and constant curvature space, or a solvmanifold. Moreover, the classification of which homogeneous space admit a non-trivial conformally Einstein metric is reduced to classifying the nilsoliton metrics. Our proof adapts Lafuente’s geometric invariant theory approach. As an application, we completely classify simply connected, irreducible, non-trivial conformally Einstein spaces in dimension five.

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