Infinite families of manifolds of positive kmathrm{th}-intermediate Ricci curvature with k small

Abstract

Positive kmathrm{th}-intermediate Ricci curvature on a Riemannian n-manifold, to be denoted by {{,mathrm{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 {{,mathrm{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 nge 7 congruent to 3,{{,mathrm{mod},}}4 supports infinitely many closed simply connected manifolds of pairwise distinct homotopy type, all of which admit homogeneous metrics of {{,mathrm{Ric},}}_k>0 for some k<n/2. We also prove that each dimension nge 4 congruent to 0 or 1,{{,mathrm{mod},}}4 supports closed manifolds which carry metrics of {{,mathrm{Ric},}}_k>0 with kle n/2, but do not admit metrics of positive sectional curvature.