Positive Ricci curvature on simply-connected manifolds with cohomogeneity-two torus actions

TL;DR

This paper provides an alternative proof that certain simply-connected manifolds with cohomogeneity-two torus actions admit metrics with positive Ricci curvature, expanding the known classes of such manifolds.

Contribution

It establishes the existence of positive Ricci curvature metrics on a broad family of simply-connected manifolds with cohomogeneity-two torus actions, correcting previous gaps in the proof.

Findings

Existence of positive Ricci curvature metrics on infinitely many simply-connected manifolds.

Application to all simply-connected 5- and 6-manifolds with torus actions.

Construction of metrics invariant under specific torus subgroups.

Abstract

A gap in the proof of the main result in reference [1] in our original submission propagated into the constructions presented in the first version of our manuscript. In this version we give an alternative proof for the existence of Riemannian metrics with positive Ricci curvature on an infinite subfamily of closed, simply-connected smooth manifolds with a cohomogeneity two torus action and recover some of our original results. Namely, we show that, for each $n\geqslant 1$, there exist infinitely many spin and non-spin diffeomorphism types of closed, smooth, simply-connected $(n+4)$-manifolds with a smooth, effective action of a torus $T^{n+2}$ and a metric of positive Ricci curvature invariant under a $T^{n}$-subgroup of $T^{n+2}$. As an application, we show that every closed, smooth, simply-connected $5$- and $6$-manifold admitting a smooth, effective torus action of cohomogeneity two supports metrics with positive Ricci curvature.