Torus actions on manifolds with positive intermediate Ricci curvature

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.