Torus actions on manifolds with positive intermediate Ricci curvature

TL;DR

This paper classifies certain high-symmetry manifolds with positive second-intermediate Ricci curvature, showing they are spheres or complex projective spaces under various conditions, and introduces new tools for studying isometric actions.

Contribution

It provides classification results for manifolds with positive second-intermediate Ricci curvature and symmetry, and develops new methods for analyzing isometric actions on such manifolds.

Findings

Odd-dimensional manifolds are spheres.

Even-dimensional manifolds (except dimension 6) have positive Euler characteristic.

Under stronger symmetry assumptions, manifolds have trivial odd cohomology and are spheres or complex projective spaces.

Abstract

We study closed, simply connected manifolds with positive $2^\mathrm{nd}$-intermediate Ricci curvature and large symmetry rank. In odd dimensions, we show that they are spheres. In even dimensions other than $6$, we show that they must have positive Euler characteristic. Under stronger assumptions on the symmetry rank, we show that such even dimensional manifolds must have trivial odd degree integral cohomology, and if the second Betti number is no more than $1$, they are either spheres or complex projective spaces. In the process, we establish new tools for studying isometric actions on closed manifolds with positive $k^\mathrm{th}$-intermediate Ricci for values of $k \geq 2$. These tools include generalizations of the isotropy rank lemma, symmetry rank bound, and connectedness principle from the setting of positive sectional curvature.