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Cohomogeneity two Bazaikin spaces

Jason DeVito, Rachel Flores

TL;DR

The paper analyzes cohomogeneity-two Bazaikin spaces equipped with Wilking metrics and shows a sharp contrast with cohomogeneity-one and homogeneous cases: the set of points with strictly positive sectional curvature does not have full measure. By identifying the unique cohomogeneity-two pattern for the defining $q$-tuple, it constructs the main Wilking metric (a single Cheeger deformation in the left factor along $S$ and a deformation along $U(4)$ on the right) and reduces curvature questions to a two-parameter family. The authors prove, via explicit construction of zero-curvature planes on an open subset, that these spaces are not almost positively curved under Wilking metrics with the corresponding isometry-action properties. The results imply that new geometric techniques are required to achieve almost-positive curvature in the cohomogeneity-two Bazaikin setting, highlighting a fundamental limitation of the Wilking construction for this class of biquotients.

Abstract

We study the sectional curvature of all of the cohomogeneity two Bazaikin spaces with respect to a Riemannian metric construction due to Wilking. We show that, in contrast to the cohomogeneity one and homogeneous case, for all of the cohomogeneity two examples, the set of points with strictly positive curvature does not have full measure.

Cohomogeneity two Bazaikin spaces

TL;DR

The paper analyzes cohomogeneity-two Bazaikin spaces equipped with Wilking metrics and shows a sharp contrast with cohomogeneity-one and homogeneous cases: the set of points with strictly positive sectional curvature does not have full measure. By identifying the unique cohomogeneity-two pattern for the defining -tuple, it constructs the main Wilking metric (a single Cheeger deformation in the left factor along and a deformation along on the right) and reduces curvature questions to a two-parameter family. The authors prove, via explicit construction of zero-curvature planes on an open subset, that these spaces are not almost positively curved under Wilking metrics with the corresponding isometry-action properties. The results imply that new geometric techniques are required to achieve almost-positive curvature in the cohomogeneity-two Bazaikin setting, highlighting a fundamental limitation of the Wilking construction for this class of biquotients.

Abstract

We study the sectional curvature of all of the cohomogeneity two Bazaikin spaces with respect to a Riemannian metric construction due to Wilking. We show that, in contrast to the cohomogeneity one and homogeneous case, for all of the cohomogeneity two examples, the set of points with strictly positive curvature does not have full measure.
Paper Structure (6 sections, 23 theorems, 54 equations)

This paper contains 6 sections, 23 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Metrics on biquotients
  3. Bazaikin spaces
  4. Background on Bazaikin spaces
  5. Metrics on Bazaikin spaces
  6. Zero-curvature planes in the main Wilking metric

Key Result

Theorem 1.1

Suppose a Bazaikin space $\mathcal{B}_{\overline{q}}$ is equipped with the Bazaikin metric. Then

Theorems & Definitions (46)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • ...and 36 more