N-Bakry Emery Ricci Curvature & N-Quasi Einstein Metrics

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 \u3e 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