Nonnegatively curved fixed point homogeneous 5-manifolds

TL;DR

This paper classifies certain 5-dimensional manifolds with nonnegative curvature that admit a specific type of symmetry action, expanding understanding of geometric structures with symmetry in low dimensions.

Contribution

It provides a classification of closed, simply connected 5-manifolds with nonnegative curvature under fixed point homogeneous group actions.

Findings

Classification of manifolds up to diffeomorphism

Identification of symmetry group actions on these manifolds

Insights into geometric structures with fixed point homogeneous actions

Abstract

Let $G$ be a compact Lie group acting effectively by isometries on a compact Riemannian manifold $M$ with nonempty fixed point set $Fix(M,G)$. We say that the action is \emph{fixed point homogeneous} if $G$ acts transitively on a normal sphere to some component of $Fix(M,G)$, equivalently, if $Fix(M,G)$ has codimension one in the orbit space of the action. We classify up to diffeomorphism closed, simply connected 5-manifolds with nonnegative sectional curvature and an effective fixed point homogeneous isometric action of a compact Lie group.