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Semi-Riemannian metrics on compact simple Lie Groups

Abdelghani Zeghib

TL;DR

The paper surveys left-invariant semi-Riemannian metrics on compact Lie groups, addressing geodesic-flow completeness, the structure of isometry and conformal groups, and the role of group extensions. It integrates classical and modern results (Marsden, OT, Chen–Liang–Zhu, BGZ) to distinguish Riemannian and non-Riemannian behavior, simple vs non-simple cases, and conformal symmetry constraints, while presenting explicit non-simple examples. A unifying theme is the use of supergroup extensions and conformal geometry to characterize symmetry and rigidity phenomena on compact groups, including maximal symmetry for Killing-form metrics and tight restrictions for essential conformal actions. The findings clarify when the isometry group must be compact, when conformal symmetry forces flat models, and how non-simple structures can yield non-compact isometry groups, informing both the geometry and potential applications in homogeneous pseudo-Riemannian geometry.

Abstract

This is a survey on left invariant semi-Riemannian metrics on compact Lie groups.

Semi-Riemannian metrics on compact simple Lie Groups

TL;DR

The paper surveys left-invariant semi-Riemannian metrics on compact Lie groups, addressing geodesic-flow completeness, the structure of isometry and conformal groups, and the role of group extensions. It integrates classical and modern results (Marsden, OT, Chen–Liang–Zhu, BGZ) to distinguish Riemannian and non-Riemannian behavior, simple vs non-simple cases, and conformal symmetry constraints, while presenting explicit non-simple examples. A unifying theme is the use of supergroup extensions and conformal geometry to characterize symmetry and rigidity phenomena on compact groups, including maximal symmetry for Killing-form metrics and tight restrictions for essential conformal actions. The findings clarify when the isometry group must be compact, when conformal symmetry forces flat models, and how non-simple structures can yield non-compact isometry groups, informing both the geometry and potential applications in homogeneous pseudo-Riemannian geometry.

Abstract

This is a survey on left invariant semi-Riemannian metrics on compact Lie groups.
Paper Structure (23 sections, 6 theorems)

This paper contains 23 sections, 6 theorems.

Table of Contents

  1. Introduction
  2. Results
  3. The geodesic flow
  4. Left Riemannian metrics
  5. Non-Riemannian case
  6. Maximally symmetric metric
  7. The conformal group
  8. Terminology: Supergroup extensions
  9. Riemannian case
  10. Conformal Group, Proof of Theorem \ref{['Conformal Theorem']}
  11. Non Riemannian case
  12. Sketch
  13. Simple case
  14. Non-simple examples
  15. Riemannian Non-simple example
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $K$ be a compact simple Lie group. Consider $g_K$, its left invariant (in fact also right invariant) metric determined by the Killing form (defined on the Lie algebra ${\mathfrak{k}}$). Then, $g_K$ is maximally symmetric among left invariant metrics, that is, for any left invariant metric $g$ on

Theorems & Definitions (7)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Theorem 4.1
  • Theorem 4.2
  • Proposition 4.3
  • proof