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 Author Pro, Curtis Title On Riemannian Submersions and Diffeomorphism Stability URL http://www.escholarship.org/uc/item/2z16d2kf Publication Date 2012 Discipline/Department Mathematics University/Publisher University of California – Riverside Abstract This thesis consists of work that was carried out in three separate papers that were written during my time at UC, Riverside. Abstract of chapter II: If $\pi:M\rightarrow B$ is a Riemannian Submersion and $M$ has non-negative sectional curvature, O'Neill's Horizontal Curvature Equation shows that $B$ must also have non-negative curvature. We find constraints on the extent to which O'Neill's horizontal curvature equation can be used to create positive curvature on the base space of a Riemannian submersion. In particular, we study when K. Tapp's theorem on Riemannian submersions of compact Lie groups with bi-invariant metrics generalizes to arbitrary manifolds of non-negative curvature.Abstract of Chapter III: Though Riemannian submersions preserve non-negative sectional curvature this does not generalize to Riemannian submersions from manifolds with non-negative Ricci curvature. We give here an example of a Riemannian submersion $\pi: M\rightarrow B$ for which $\textrm{Ricci}_p(M)>0$ and at some point $p\in B$, $\text{Ricci}_p(B)<0$. Abstract of Chapter IV: The smallest $r$ so that a metric $r$ – ball covers a metric space $M$ is called the radius of $M.$ The volume of a metric $r$-ball in the space form of constant curvature $k$ is an upper bound for the volume of any Riemannian manifold with sectional curvature $\geq k$ and radius $\leq r$. We show that when such a manifold has volume almost equal to this upper bound, it is diffeomorphic to a sphere or a real projective space. Subjects/Keywords Mathematics; Diffeomorphsim Stability; Dual Foliations; Isometric Group Actions; Ricci Curvature; Riemannian Submersions Language en Rights public Country of Publication us Format application/pdf Record ID california:qt2z16d2kf Other Identifiers qt2z16d2kf Repository california Date Indexed 2018-02-26

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…volume almost equal to this upper bound, it is diffeomorphic to a sphere or a real projective space. vii Contents 1 Introduction 1.1 Flats and Submersions in Non-Negative Curvature . . . . . . . . . . . . . . 1.2 Riemannian Submersions Need Not…

…2.3 Jacobi Fields Along Geodesics Contained In Flats . . 2.4 The Holonomy of π . . . . . . . . . . . . . . . . . . . 1 1 5 6 . . . . 11 11 12 14 20 3 Riemannian Submersions Need Not Preserve Positive Curvature 3.1 Vertical Warping…

…into the sum of two non-negative quantities, we see immediately that Riemannian submersions preserve nonnegative curvature. In addition, if either term on the right is positive, then secB (x, y) > 0. Naively, one might expect positively curved…

…curved metrics so that π and Q : P × F → P ×G F = E become Riemannian submersions. If σ ˜ is a π–horizontal zero-curvature plane in E such that expp (˜ σ ) is a flat, then 1 σ ˜ projects to a zero-curvature plane σ in B that exponentiates to a…

…Theorems 2, 3, and 4. The associated bundles give examples of Theorem 2 and Corollary 5. 1.2 Riemannian Submersions Need Not Preserve Positive Ricci Curvature One might ask if something similar to O’Neill’s horizontal curvature equation exists for…

Riemannian submersions in the Ricci curvature case. However, given the difference between Ricci and sectional curvature, it is not a surprise that Riemannian submersions need not preserve a lower Ricci cuvature bound. Yet, an example of this appears to be…

…is to generalize Tapp’s theorem to Riemannian submersions from more than just compact Lie groups G with a biinvariant metric. So we begin by giving examples of how the conclusions of Tapp’s theorem can fail to hold in this new setting. Recall that…

Riemannian submersions of compact, nonnegatively curved manifolds. 13 Example 16 To see how conclusion 1 of Tapp’s theorem can fail to hold, choose ψ in the previous example to be constant in a neighborhood of π/2. This makes S 2 , gψ isometric to a flat…