Nonnegatively curved fixed point homogeneous manifolds in low dimensions

TL;DR

This paper classifies low-dimensional fixed-point homogeneous manifolds with nonnegative curvature, extending known classifications from positive curvature to include new cases in dimensions 3 and 4.

Contribution

It provides a complete classification of fixed-point homogeneous manifolds in dimensions 3 and 4 with nonnegative curvature, including analysis of circle actions on simply-connected 4-manifolds.

Findings

Classified fixed-point homogeneous manifolds in dimensions 3 and 4.

Determined which simply-connected 4-manifolds admit specific circle actions.

Extended classification from positive to nonnegative curvature cases.

Abstract

Let $G$ be a compact Lie group acting isometrically on a compact Riemannian manifold $M$ with nonempty fixed point set $M^G$. We say that $M$ is fixed-point homogeneous if $G$ acts transitively on a normal sphere to some component of $M^G$. Fixed-point homogeneous manifolds with positive sectional curvature have been completely classified. We classify fixed-point homogeneous Riemannian manifolds in dimensions 3 and 4 and determine which nonnegatively curved simply-connected 4-manifolds admit a smooth fixed-point homogeneous circle action with a given orbit space structure.