Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Rigidity of Lie foliations with locally symmetric leaves

Gael Meigniez, Hiraku Nozawa

Abstract

We prove that if the leaves of a minimal Lie foliation are locally isometric to a symmetric space of non-compact type without a Poincare disk factor, then the foliation is smoothly conjugate to a homogeneous Lie foliation up to finite covering. This result generalizes and strengthens Zimmer's theorem, which characterizes minimal Lie foliations with leaves isometric to a symmetric space of non-compact type without real rank one factors as pullbacks of homogeneous foliations. As applications, we extend Zimmer's arithmeticity theorem for holonomy groups and establish a rigidity theorem for Riemannian foliations with locally symmetric leaves.

Rigidity of Lie foliations with locally symmetric leaves

Abstract

We prove that if the leaves of a minimal Lie foliation are locally isometric to a symmetric space of non-compact type without a Poincare disk factor, then the foliation is smoothly conjugate to a homogeneous Lie foliation up to finite covering. This result generalizes and strengthens Zimmer's theorem, which characterizes minimal Lie foliations with leaves isometric to a symmetric space of non-compact type without real rank one factors as pullbacks of homogeneous foliations. As applications, we extend Zimmer's arithmeticity theorem for holonomy groups and establish a rigidity theorem for Riemannian foliations with locally symmetric leaves.

Paper Structure

This paper contains 10 sections, 17 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Main result and corollaries
  3. Quasi-action of the structural group
  4. Rigidity of Lie foliations with hyperbolic leaves
  5. Rigidity of Lie foliations with leaves locally isometric to an irreducible symmetric space of higher rank
  6. Rigidity of Lie foliations with locally symmetric leaves
  7. Quasi-isometric rigidity of Lie foliations
  8. Homogeneous Lie foliations
  9. Application to Riemannian foliations
  10. Problems

Key Result

Proposition 2.1

Let $M$ be a closed connected manifold; denote by $\widetilde{M}$ its universal cover. A foliation $\mathcal{F}$ on $M$ is a $G$-Lie foliation if and only if one has

Theorems & Definitions (35)

  • Proposition 2.1: Fedida
  • Example 2.2: Homogeneous Lie foliations
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.6
  • Example 2.7: Suspension Lie foliations
  • Corollary 2.8
  • Corollary 2.9
  • Corollary 2.10
  • ...and 25 more