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Collapsed limits of compact Heisenberg manifolds with sub-Riemannian metrics

Kenshiro Tashiro

Abstract

In this paper, we show that every collapsed Gromov--Hausdorff limit of compact Heisenberg manifolds is isometric to a flat torus. Here we say that a sequence of sub-Riemannian manifolds collapses if their total measure with respect to the Popp's volume or the minimal Popp's volume converges to zero. In the appendix, we give the systolic inequality on sub-Riemannian Heisenberg manifolds, and observe that the exponent of the total measure is equal to the inverse of the Hausdorff dimension.

Collapsed limits of compact Heisenberg manifolds with sub-Riemannian metrics

Abstract

In this paper, we show that every collapsed Gromov--Hausdorff limit of compact Heisenberg manifolds is isometric to a flat torus. Here we say that a sequence of sub-Riemannian manifolds collapses if their total measure with respect to the Popp's volume or the minimal Popp's volume converges to zero. In the appendix, we give the systolic inequality on sub-Riemannian Heisenberg manifolds, and observe that the exponent of the total measure is equal to the inverse of the Hausdorff dimension.

Paper Structure

This paper contains 14 sections, 20 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Preliminaries from sub-Riemannian geometry
  3. Sub-Riemannian structure
  4. Length minimizing paths
  5. The Popp's volume
  6. The minimal Popp's volume form on Lie groups
  7. Compact Heisenberg manifolds
  8. The moduli space of compact Heisenberg manifolds
  9. The operator $j$
  10. Geodesics
  11. Volume forms
  12. The circle bundle structure
  13. The conditions for fibers collapsing to a point
  14. Appendix

Key Result

Theorem \oldthetheorem

Let $\left\{(\Gamma_k\backslash H_n,dist_k)\right\}_{k\in\mathbb{N}}$ be a sequence of compact Heisenberg manifolds endowed with left invariant sub-Riemannian metrics. Assume that this sequence converges in the Gromov--Hausdorff topology, the diameter is uniformly bounded above by $D>0$, and the tot

Theorems & Definitions (42)

  • Theorem \oldthetheorem: Main result
  • Corollary 1.1
  • Example 2.1
  • Definition 2.1: Bracket generating distribution
  • Theorem \oldthetheorem: Chow--Rashevskii's theorem, Theorem 3.31 in agr
  • Theorem \oldthetheorem: The Pontryagin maximal principle, Theorem 4.20 in agr
  • Definition 2.2: Normal extremal
  • Definition 2.3: Abnormal extremal
  • Example 2.2
  • Remark 2.1
  • ...and 32 more