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

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 $n⩾1$, 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.

11 pages, to appear in Proceedings of the American Mathematical Society

References in corpus (3)

• The classification of compact simply connected biquotients in dimension 6 and 7
• How to lift positive Ricci curvature
• Rational Ellipticity in Cohomogeneity Two

Cited by in corpus (1)

• Generalized Surgery on Riemannian Manifolds of Positive Ricci Curvature