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Einstein Lie groups, geodesic orbit manifolds and regular Lie subgroups

Nikolaos Panagiotis Souris

Abstract

We study the relation between two special classes of Riemannian Lie groups $G$ with a left-invariant metric $g$: The Einstein Lie groups, defined by the condition $\operatorname{Ric}_g=cg$, and the geodesic orbit Lie groups, defined by the property that any geodesic is the integral curve of a Killing vector field. The main results imply that extensive classes of compact simple Einstein Lie groups $(G,g)$ are not geodesic orbit manifolds, thus providing large-scale answers to a relevant question of Y. Nikonorov. Our approach involves studying and characterizing the $G\times K$-invariant geodesic orbit metrics on Lie groups $G$ for a wide class of subgroups $K$ that we call (weakly) regular. By-products of our work are structural and characterization results that are of independent interest for the classification problem of geodesic orbit manifolds.

Einstein Lie groups, geodesic orbit manifolds and regular Lie subgroups

Abstract

We study the relation between two special classes of Riemannian Lie groups with a left-invariant metric : The Einstein Lie groups, defined by the condition , and the geodesic orbit Lie groups, defined by the property that any geodesic is the integral curve of a Killing vector field. The main results imply that extensive classes of compact simple Einstein Lie groups are not geodesic orbit manifolds, thus providing large-scale answers to a relevant question of Y. Nikonorov. Our approach involves studying and characterizing the -invariant geodesic orbit metrics on Lie groups for a wide class of subgroups that we call (weakly) regular. By-products of our work are structural and characterization results that are of independent interest for the classification problem of geodesic orbit manifolds.

Paper Structure

This paper contains 16 sections, 34 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Overview of the proof and additional characterization results
  3. Structure of the paper
  4. Preliminaries on invariant metrics on Lie groups and homogeneous spaces
  5. Left-invariant and $G\times K$-invariant metrics
  6. The general form of a $G\times K$-invariant metric on $G$
  7. Invariant metrics on homogeneous spaces
  8. Geodesic orbit manifolds
  9. Geodesic orbit manifolds and naturally reductive manifolds
  10. Riemannian homogeneous spaces and Lie groups as geodesic orbit manifolds
  11. Regular subgroups and their properties
  12. Proof of Theorem \ref{['consequence']}
  13. Some preliminary results
  14. Proof of Theorem \ref{['consequence']}
  15. Applications of Theorem \ref{['consequence']}
  16. ...and 1 more sections

Key Result

Theorem 1.4

Let $(G,g)$ be a connected compact Riemannian simple Lie group and assume that the metric $g$ is $G\times K$-invariant, where $K$ is a closed regular subgroup of $G$. If $(G,g)$ is a $G\times K$-geodesic orbit manifold then $(G,g)$ is a naturally reductive manifold.

Theorems & Definitions (55)

  • Definition 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Corollary 1.9
  • Theorem 1.10
  • Lemma 1.11
  • Corollary 1.12
  • ...and 45 more