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Metric Lie Groups. Carnot-Carathéodory spaces from the homogeneous viewpoint

Enrico Le Donne

Abstract

This book explores geometries defined by left-invariant distance functions on Lie groups, with a particular focus on nilpotent groups and Carnot groups equipped with geodesic distances. Geodesic left-invariant metrics are either sub-Riemannian or their generalizations, known as sub-Finsler geometries or Carnot-Carathéodory metrics. The primary objective is to illustrate how these non-smooth geometries, together with a Lie group structure, manifest in various mathematical fields, including metric geometry and geometric group theory. Additionally, the book demonstrates the role of metric Lie groups, particularly Carnot groups, in the following contexts: (a) as asymptotic cones of nilpotent groups; (b) as parabolic boundaries of rank-one symmetric spaces and, more broadly, of homogeneous negatively curved Riemannian manifolds; (c) as limits of Riemannian manifolds and tangents of sub-Riemannian manifolds.

Metric Lie Groups. Carnot-Carathéodory spaces from the homogeneous viewpoint

Abstract

This book explores geometries defined by left-invariant distance functions on Lie groups, with a particular focus on nilpotent groups and Carnot groups equipped with geodesic distances. Geodesic left-invariant metrics are either sub-Riemannian or their generalizations, known as sub-Finsler geometries or Carnot-Carathéodory metrics. The primary objective is to illustrate how these non-smooth geometries, together with a Lie group structure, manifest in various mathematical fields, including metric geometry and geometric group theory. Additionally, the book demonstrates the role of metric Lie groups, particularly Carnot groups, in the following contexts: (a) as asymptotic cones of nilpotent groups; (b) as parabolic boundaries of rank-one symmetric spaces and, more broadly, of homogeneous negatively curved Riemannian manifolds; (c) as limits of Riemannian manifolds and tangents of sub-Riemannian manifolds.

Paper Structure

This paper contains 295 sections, 240 theorems, 1231 equations, 27 figures.

Table of Contents

  1. Introduction
  2. What sub-Riemannian geometry is
  3. Content and structure of this text
  4. Sub-Riemannian geometries as models
  5. Examples from mathematics
  6. Control theory and metric geometry
  7. Geometric group theory and asymptotic geometry
  8. Complex analysis and Cauchy-Riemann geometry
  9. Analysis of hypoelliptic operators and singular integral operators
  10. Classical mechanics and non-holonomic dynamical systems
  11. Contact geometry, Engel structures, and exterior differential systems
  12. Riemannian geometry
  13. Univalent function theory
  14. Diffusion processes and financial mathematics
  15. Examples from physics
  16. ...and 280 more sections

Key Result

Theorem A

The sub-Finsler Carnot groups are the only metric spaces that are locally compact, geodesic, isometrically homogeneous, and admitting metric dilations.

Figures (27)

  • Figure 1: Interdependence of chapters. Thick boxes represent the main chapters. Dashed boxes represent chapters devoted to examples.
  • Figure 2: A contact distribution on ${\mathbb R}^3$ is a polarization by planes.
  • Figure 3: A Carnot-Carathéodory ball, which is the limit of large balls for a word distance on a finitely generated nilpotent group. See Example \ref{['example:standard generating set in Heisenberg']}.
  • Figure 4: The cat spins itself around and right itself.
  • Figure 5: A ball rolling on the plane without sliding, nor spinning.
  • ...and 22 more figures

Theorems & Definitions (570)

  • Theorem A
  • Theorem B: Pansu's Rademacher Theorem
  • Definition 1.2.3: Standard contact form
  • Proposition 1.3.5
  • Lemma 1.4.5
  • Corollary 1.4.7
  • Corollary 1.4.9
  • proof
  • Corollary 1.4.11
  • Proposition 1.4.14
  • ...and 560 more