Full Record

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 |

Sample Search Hits | Sample Images

…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 . . . . . . . . . . . . . . . . . . .
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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…