Low Dimensional Polar Actions

TL;DR

This paper classifies low-dimensional compact simply-connected polar manifolds with group actions, and identifies which admit metrics with non-negative curvature, advancing understanding of symmetric geometric structures.

Contribution

It provides a complete equivariant classification of polar manifolds in dimensions 5 or less and characterizes those with non-negative curvature metrics.

Findings

Complete classification of polar manifolds in dimension ≤5

Identification of polar actions admitting non-negative curvature

Advancement in understanding symmetric Riemannian manifolds

Abstract

Polar manifolds are Riemannian G-manifolds admitting a "section", i.e., a complete submanifold passing through every orbit and doing so orthogonally. We consider compact simply-connected polar manifolds and achieve an equivariantly diffeomorphic classification in dimensions 5 or less. As an application, we determine which of these polar actions admit an invariant metric with non-negative curvature.