GKM actions on cohomogeneity one manifolds

TL;DR

This paper characterizes when cohomogeneity one manifolds with a compact Lie group action admit GKM-type torus actions, providing conditions and describing the associated GKM graphs with explicit examples.

Contribution

It establishes necessary and sufficient conditions for GKM actions on cohomogeneity one manifolds and describes their GKM graphs, expanding understanding of symmetry actions on these manifolds.

Findings

Derived conditions for GKM actions on cohomogeneity one manifolds.

Described GKM graphs explicitly for various examples.

Connected group diagrams to GKM properties.

Abstract

We consider compact manifolds $M$ with a cohomogeneity one action of a compact Lie group $G$ such that the orbit space $M/G$ is a closed interval. For $T$ a maximal torus of $G$, we find necessary and sufficient conditions on the group diagram of $M$ such that the $T$-action on $M$ is of GKM type, and describe its GKM graph. The general results are illustrated on explicit examples.