Cohomogeneity-two torus actions on non-negatively curved manifolds of low dimension

TL;DR

This paper classifies certain torus actions on low-dimensional, non-negatively curved manifolds, showing they are equivalent to isometric actions on normal biquotients, thus advancing understanding of symmetry in geometric topology.

Contribution

It proves that all effective torus actions on specific low-dimensional, non-negatively curved manifolds are diffeomorphic to isometric actions on normal biquotients, providing a classification result.

Findings

Any smooth, effective $T^{n-2}$ action on $M^n$ is diffeomorphic to an isometric action on a normal biquotient.

Effective circle actions on 4-manifolds are also diffeomorphic to isometric actions on normal biquotients.

The results hold for manifolds with dimensions 4, 5, and 6, under the given curvature and symmetry conditions.

Abstract

Let $M^n$, $n \in \{4,5,6\}$, be a compact, simply connected $n$-manifold which admits some Riemannian metric with non-negative curvature and an isometry group of maximal possible rank. Then any smooth, effective action on $M^n$ by a torus $T^{n-2}$ is equivariantly diffeomorphic to an isometric action on a normal biquotient. Furthermore, it follows that any effective, isometric circle action on a compact, simply connected, non-negatively curved four-dimensional manifold is equivariantly diffeomorphic to an effective, isometric action on a normal biquotient.