UC Riverside

This dissertation studies two topics in Riemannian geometry.First, we study the existence of totally geodesic submanifolds in Riemannian 3-manifolds. Murphy and Wilhelm showed that a generic closed Riemannian manifold has no totally geodesic submanifolds, provided the ambient space is at least four dimensional. We show that the set of metrics that admit totally geodesic submanifolds on a compact 3-manifold actually contains a set that is open and dense set in the Cq -topology, provided q ≥ 3.Second, we study the preservation of positive intermediate Ricci curvature under Riemannian submersions. Pro and Wilhelm showed that there are Riemannian submersions π : M → B with M a compact manifold with positive Ricci curvature, whose b-dimensional base has Ricci curvatures with both signs. We show that if Rick(M) > 0, then Rick(B) must be positive if k ∈ {1, 2, · · · , b − 1}, yet Ric(B) need not be positive if k = b.