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Non-singular geodesic orbit nilmanifolds

Yuri Nikolayevsky, Wolfgang Ziller

Abstract

A Riemannian manifold is called a geodesic orbit manifolds, GO for short, if any geodesic is an orbit of a one-parameter group of isometries. By a result of C.Gordon, a non-flat GO nilmanifold is necessarily a two-step nilpotent Lie group with a left-invariant metric. We give a complete classification of non-singular GO nilmanifolds. Besides previously known examples, there are new families with 3-dimensional center, and two one-parameter families of dimensions 14 and 15.

Non-singular geodesic orbit nilmanifolds

Abstract

A Riemannian manifold is called a geodesic orbit manifolds, GO for short, if any geodesic is an orbit of a one-parameter group of isometries. By a result of C.Gordon, a non-flat GO nilmanifold is necessarily a two-step nilpotent Lie group with a left-invariant metric. We give a complete classification of non-singular GO nilmanifolds. Besides previously known examples, there are new families with 3-dimensional center, and two one-parameter families of dimensions 14 and 15.

Paper Structure

This paper contains 27 sections, 22 theorems, 18 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries. Structure of GO-pairs
  3. Normalizer, centralizer and reducibility
  4. The GO condition
  5. GO subspaces of centralizer type
  6. GO-pairs of Clifford type
  7. Non-singular GO-pairs of Rep type
  8. Examples
  9. $dim V = 3$
  10. $dim \unichar{"1D51E} = 8$
  11. Proof of Theorem \ref{['th:class']} for $dim \unichar{"1D51E} \unichar{"2264} 8$
  12. Proof of Theorem \ref{['th:class']}
  13. Preparation
  14. Dimension bounds
  15. Proof of Proposition \ref{['p:irst']} for $dim \unichar{"1D51E} > 8$
  16. ...and 12 more sections

Key Result

Theorem A

Any geodesic orbit nilmanifold is either two-step or abelian.

Theorems & Definitions (43)

  • Theorem A: Gor
  • Definition
  • Theorem B: Gor
  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Definition
  • Remark 1
  • ...and 33 more