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Failure of Lichnerowicz-type result in parabolic geometries of real rank at least $3$

Toby Aldape

Abstract

Given a Yamaguchi nonrigid parabolic model geometry $(G,P)$ with $G$ simple of real rank at least $3$, we use techniques developed by Erickson to establish the existence of closed, nonflat, essential, regular, normal Cartan geometries modeled on $(G,P)$. Yamaguchi nonrigidity is a necessary condition for admitting nonflat, regular, normal examples. This rules out Lichnerowicz-type conjectures for these model geometries.

Failure of Lichnerowicz-type result in parabolic geometries of real rank at least $3$

Abstract

Given a Yamaguchi nonrigid parabolic model geometry with simple of real rank at least , we use techniques developed by Erickson to establish the existence of closed, nonflat, essential, regular, normal Cartan geometries modeled on . Yamaguchi nonrigidity is a necessary condition for admitting nonflat, regular, normal examples. This rules out Lichnerowicz-type conjectures for these model geometries.

Paper Structure

This paper contains 16 sections, 37 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Notation
  4. Structure Theory
  5. Real and Complex Representations
  6. Harmonic Curvature
  7. Borel-Weil-Bott
  8. Scaling Elements
  9. Curvature Trees and Harmonic Seeds
  10. Compact Quotients of Curvature Trees
  11. Lowest Weights
  12. Case Analysis
  13. Non-scaling Weights
  14. Existence of $a_0$
  15. The case $(\sl_4(\mathbb{H}),P_{2,6})$
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $(M,[g])$ be a connected, essential, Riemannian conformal manifold. Then $M$ is conformally diffeomorphic to either the round sphere or Euclidean space.

Theorems & Definitions (66)

  • Theorem 1.1: Ferrand-Obata
  • Theorem 1.2: Schoen-Webster
  • Theorem 1.3
  • Theorem 1.4: Main Theorem
  • Conjecture 1.5: Lorentzian Lichnerowicz
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • ...and 56 more