Infinite families of manifolds of positive kth-intermediate Ricci curvature with k small

Abstract

Positive kth-intermediate Ricci curvature on a Riemannian n-manifold, to be denoted by Ric_k > 0, is a condition that interpolates between positive sectional and positive Ricci curvature (when k = 1 and k = n − 1 respectively). In this work, we produce many examples of manifolds of Ric_k > 0 with k small by examining symmetric and normal homogeneous spaces, along with certain metric deformations of fat homogeneous bundles. As a consequence, we show that every dimension n ≥ 7 congruent to 3 mod 4 supports infinitely many closed simply connected manifolds of pairwise distinct homotopy type, all of which admit homogeneous metrics of Ric_k > 0 for some k < n/2. We also prove that each dimension n ≥ 4 congruent to 0 or 1 mod 4 supports closed manifolds which carry metrics of Ric_k > 0 with k ≤ n/2, but do not admit metrics of positive sectional curvature.