Infinite families of manifolds of positive $$k\mathrm{th}$$-intermediate Ricci curvature with k small

Abstract

AbstractPositive$$k\mathrm{th}$$kth-intermediate Ricci curvature on a Riemanniann-manifold, to be denoted by$${{\,\mathrm{Ric}\,}}_k>0$$Rick>0, is a condition that interpolates between positive sectional and positive Ricci curvature (when$$k =1$$k=1and$$k=n-1$$k=n-1respectively). In this work, we produce many examples of manifolds of$${{\,\mathrm{Ric}\,}}_k>0$$Rick>0withksmall by examining symmetric and normal homogeneous spaces, along with certain metric deformations of fat homogeneous bundles. As a consequence, we show that every dimension$$n\ge 7$$n≥7congruent to$$3\,{{\,\mathrm{mod}\,}}4$$3mod4supports infinitely many closed simply connected manifolds of pairwise distinct homotopy type, all of which admit homogeneous metrics of$${{\,\mathrm{Ric}\,}}_k>0$$Rick>0for some$$k<n/2$$k<n/2. We also prove that each dimension$$n\ge 4$$n≥4congruent to 0 or$$1\,{{\,\mathrm{mod}\,}}4$$1mod4supports closed manifolds which carry metrics of$${{\,\mathrm{Ric}\,}}_k>0$$Rick>0with$$k\le n/2$$k≤n/2, but do not admit metrics of positive sectional curvature.