#### Date of Award

Spring 5-22-2021

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### Advisor(s)

William Wylie

#### Keywords

Differential Geometry, Global Analysis, Manifolds, N-Bakry Emery Ricci, N-quasi Einstein, Riemannian Geometry

#### Subject Categories

Physical Sciences and Mathematics

#### Abstract

We begin the thesis by giving an intuitive introduction to calculus on mani- folds for the non-mathematician. We then give a semi-intuitive description on Ricci curvature for the non-geometer. We give a description of the N-Bakry- Émery Ricci curvature and the N-quasi Einstein metric. The main results in this thesis are related to the N-Bakry-Émery Ricci curvature and the N-quasi Einstein metric.

Our first set of main results are as follows. We generalize topological results known for noncompact manifolds with nonnegative Ricci curvature to spaces with nonnegative N-Bakry Émery Ricci curvature. We study the Splitting Theorem and a property called the geodesic loops to infinity property in relation to spaces with nonnegative N-Bakry Émery Ricci Curvature. In addition, we show that if M^n is a complete, noncompact Riemannian manifold with non- negative N-Bakry Émery Ricci curvature where N > n, then Hn-1(M,Z) is 0.

For our second set of main results, we classify the compact locally homogeneous non-gradient N-quasi Einstein 3-manifolds. Along the way, we also prove that given a compact quotient of a Lie group of any dimension that is N-quasi Einstein, the potential vector field X must be left invariant and Killing. We also classify the nontrivial N-quasi Einstein metrics that are a compact quotient of be the product of two Einstein metrics. We also show that S^1 is the only compact manifold of any dimension which admits a metric which is nontrivially N-quasi Einstein and Einstein.

#### Access

Open Access

#### Recommended Citation

Lim, Alice Wu, "N-bakry Emery Ricci Curvature & N-quasi Einstein Metrics" (2021). *Dissertations - ALL*. 1318.

https://surface.syr.edu/etd/1318