On different notions of positivity of curvature

Doctoral Dissertation

Abstract

We study interactions between the geometry and topology of Riemannian manifolds that satisfy curvature positivity conditions closely related to positive sectional curvature (sec>0). First, we discuss two notions of weakly positive curvature, defined in terms of averages of pairs of sectional curvatures. The manifold S^2 x S^2 is proved to satisfy these curvature positivity conditions, implying it satisfies a property intermediate between sec>0 and positive Ricci curvature, and between sec>0 and nonnegative sectional curvature. Combined with surgery techniques, this construction allows to classify (up to homeomorphism) the closed simply-connected 4-manifolds that admit a Riemannian metric for which averages of pairs of sectional curvatures of orthogonal planes are positive. Second, we study the notion of strongly positive curvature, which is intermediate between sec>0 and positive-definiteness of the curvature operator. We elaborate on joint work with Mendes, which yields the classification of simply-connected homogeneous spaces that admit an invariant metric with strongly positive curvature. These methods are then used to study the moduli space of homogeneous metrics with strongly positive curvature on the Wallach flag manifolds and on Berger spheres.

Attributes

Attribute NameValues
URN
  • etd-04092015-111925

Author Renato G. Bettiol
Advisor Karsten Grove
Contributor Gabor Szekelyhidi, Committee Member
Contributor Gerard K. Misiolek, Committee Member
Contributor Karsten Grove, Committee Chair
Contributor Frederico J. Xavier, Committee Member
Degree Level Doctoral Dissertation
Degree Discipline Mathematics
Degree Name PhD
Defense Date
  • 2015-03-26

Submission Date 2015-04-09
Country
  • United States of America

Subject
  • Curvature

  • 4-manifolds

  • Riemannian geometry

  • Sectional curvature

  • Homogeneous spaces

  • Curvature operator

  • Metric deformations

Publisher
  • University of Notre Dame

Language
  • English

Record Visibility and Access Public
Content License
  • All rights reserved

Departments and Units

Files

Please Note: You may encounter a delay before a download begins. Large or infrequently accessed files can take several minutes to retrieve from our archival storage system.