Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Positive curvature and discrete abelian symmetry

Lee Kennard, Elahe Khalili Samani, Catherine Searle

Abstract

By replacing the torus with an elementary abelian two-group, we generalize the maximal symmetry result of Grove and Searle and the half-maximal symmetry result of Wilking for positively curved manifolds with an isometric torus action.

Positive curvature and discrete abelian symmetry

Abstract

By replacing the torus with an elementary abelian two-group, we generalize the maximal symmetry result of Grove and Searle and the half-maximal symmetry result of Wilking for positively curved manifolds with an isometric torus action.

Paper Structure

This paper contains 18 sections, 22 theorems, 44 equations, 2 tables.

Table of Contents

  1. Estimates for error correcting codes
  2. Results from the theory of transformation groups
  3. Applications of the Connectedness Lemma
  4. Wilking's Connectedness and Periodicity Lemmas
  5. Lifting periodicity
  6. From periodic cohomology to homotopy type
  7. The Borel Formula
  8. The Small Codimension Lemmas
  9. Proof of Theorem \ref{['thm:n']}
  10. Proof of Theorem \ref{['thm:nover2']}
  11. Fundamental Group Results
  12. The Proof of Theorem \ref{['thm:nover2']}
  13. Proof of Theorem \ref{['thm:nover4']}
  14. Examples
  15. The Maximal ${\bf \mathbb{Z}_2}$-Fixed Point Symmetry Rank Case
  16. ...and 3 more sections

Key Result

Theorem A

Let $M^n$ be a closed, positively curved manifold such that $\mathbb{Z}_2^r$ acts isometrically on $M$ with a non-empty fixed-point set. If $r \geq n$, then $r=n$ and $M$ is equivariantly diffeomorphic to $S^r$ or $\mathbb{R}\mathrm{P}^r$ with a linear $\mathbb{Z}_2^r$-action.

Theorems & Definitions (52)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Definition 1.1: Hamming weight
  • Theorem 1.2
  • Lemma 1.3
  • proof : Sketch of the Proof of Lemma \ref{['ECC']}
  • proof : Proof of Theorem \ref{['thm:ECCWilking']}
  • Lemma 1.4
  • ...and 42 more