Low-dimensional manifolds with non-negative curvature and maximal symmetry rank

TL;DR

This paper classifies certain high-symmetry, non-negatively curved manifolds in low dimensions, revealing unique structures in specific cases and identifying them as products of 3-spheres.

Contribution

It provides a complete classification of closed, simply connected non-negatively curved manifolds with maximal symmetry rank in dimensions 2 to 6, highlighting unique diffeomorphism types.

Findings

In dimensions 3 and 6, the manifolds are diffeomorphic to products of 3-spheres.

The classification is complete for dimensions 2 to 6.

There is a unique such manifold in each dimension divisible by 3, corresponding to the product of 3-spheres.

Abstract

We classify closed, simply connected $n$-manifolds of non-negative sectional curvature admitting an isometric torus action of maximal symmetry rank in dimensions $2\leq n\leq 6$. In dimensions $3k$, $k=1,2$ there is only one such manifold and it is diffeomorphic to the product of $k$ copies of the 3-sphere.