Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Actions on positively curved manifolds and boundary in the orbit space

Claudio Gorodski, Andreas Kollross, Burkhard Wilking

Abstract

We study isometric actions of compact Lie groups on complete orientable positively curved $n$-manifolds whose orbit spaces have non-empty boundary in the sense of Alexandrov geometry. In particular, we classify quotients of the unit sphere by actions of compact simple Lie groups with non-empty boundary. We deduce from this the list of representations of compact simple Lie groups that admit non-trivial reductions. As a tool of special interest, we introduce a new geometric invariant of a compact symmetric space, namely, the minimal number of points in a "spanning set" of the space.

Actions on positively curved manifolds and boundary in the orbit space

Abstract

We study isometric actions of compact Lie groups on complete orientable positively curved -manifolds whose orbit spaces have non-empty boundary in the sense of Alexandrov geometry. In particular, we classify quotients of the unit sphere by actions of compact simple Lie groups with non-empty boundary. We deduce from this the list of representations of compact simple Lie groups that admit non-trivial reductions. As a tool of special interest, we introduce a new geometric invariant of a compact symmetric space, namely, the minimal number of points in a "spanning set" of the space.
Paper Structure (24 sections, 18 theorems, 33 equations, 5 tables)

This paper contains 24 sections, 18 theorems, 33 equations, 5 tables.

Table of Contents

  1. Introduction
  2. General observations
  3. The case of quotients of the sphere
  4. The complexity of orbit spaces
  5. Quaternionic representations
  6. Dimension estimate
  7. Outline of proof of Theorem \ref{['main']}
  8. The Abelian case
  9. Example
  10. Structure of the paper
  11. Preliminaries
  12. Nice involutions
  13. Generic totally geodesics submanifolds of compact symmetric spaces
  14. Proof of Theorem \ref{['simple']}
  15. Proof of the main result
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $G$ be a compact connected simple Lie group. Then there is an explicit, positive integer $\mathcal{L}_G$, depending only on the local isomorphism class of $G$, such that: For every effective and isometric action of $G$ on a connected complete orientable Riemannian manifold $M$ of positive sectio

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Lemma 3.1
  • Remark 3.2
  • Lemma 4.1
  • ...and 11 more