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1. Pro, Curtis. On Riemannian Submersions and Diffeomorphism Stability.

Degree: Mathematics, 2012, University of California – Riverside

URL: http://www.escholarship.org/uc/item/2z16d2kf

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 π:M → 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 π: M → B for which {Ricci}_{p}(M)>0 and at some point p∈ B, {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 ≥ k and radius ≤ 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

…metric foliation F the so-called *dual* foliation
F # . The *dual* leaf through a point p ∈ M is… …Let L#
p be the *dual* leaf through p.
−1
We shall see that for any p ∈ M, hol(b) is… …Since F is given by the orbit decomposition of an isometric group action, the *dual*
foliation…

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APA (6^{th} Edition):

Pro, C. (2012). On Riemannian Submersions and Diffeomorphism Stability. (Thesis). University of California – Riverside. Retrieved from http://www.escholarship.org/uc/item/2z16d2kf

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Pro, Curtis. “On Riemannian Submersions and Diffeomorphism Stability.” 2012. Thesis, University of California – Riverside. Accessed December 02, 2020. http://www.escholarship.org/uc/item/2z16d2kf.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Pro, Curtis. “On Riemannian Submersions and Diffeomorphism Stability.” 2012. Web. 02 Dec 2020.

Vancouver:

Pro C. On Riemannian Submersions and Diffeomorphism Stability. [Internet] [Thesis]. University of California – Riverside; 2012. [cited 2020 Dec 02]. Available from: http://www.escholarship.org/uc/item/2z16d2kf.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Pro C. On Riemannian Submersions and Diffeomorphism Stability. [Thesis]. University of California – Riverside; 2012. Available from: http://www.escholarship.org/uc/item/2z16d2kf

Not specified: Masters Thesis or Doctoral Dissertation