Polar manifolds and actions

TL;DR

This paper introduces a method to construct manifolds with polar group actions by specifying isotropy groups, expanding the understanding of such actions beyond symmetric spaces and exploring their topological properties.

Contribution

It generalizes classical constructions for cohomogeneity manifolds to a broader class of polar actions, providing new examples and topological insights.

Findings

Constructed manifolds with polar actions from prescribed isotropy groups.

Demonstrated the richness of polar actions through numerous examples.

Linked the topology of sections to the topology of the manifolds.

Abstract

A group action is called polar if there exists an immersed submanifold (a section) which intersects all orbits orthogonally. Such group actions have been studied extensively on symmetric spaces. We show how to construct a manifold admitting a polar group action by prescribing their isotropy groups along a fundamental domain, generalizing the classical construction for cohomogeneity manifolds. We give many examples showing the richness of this class of group actions. We also relate the topology of the section to the topology of the manifold. This is a replacement of an earlier version. Small changes, and a correction in Lemma 2.4.