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

Abstract

Positive $k^{\mathrm{th}}$-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 $n\ge 7$ 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 $n\ge 4$ congruent to $0$ or $1\mathrm{mod}4$ supports closed manifolds which carry metrics of ${\mathrm{Ric}}_k>0$ with $k\le n/2$, but do not admit metrics of positive sectional curvature.

V2. Major revision. Former Lemma 4.15 was incorrect, and consequently so were former Thms 4.8 and G. Former Thms A and B were incorrect in some dimensions, and are now subsumed into a corrected version of Thm A. The other main results (previous Thms C,D,E and F) remain unchanged. V2 includes new results (Thms B and E) on the Ric_k of non-simply connected manifolds and products of Lie groups

References in corpus (1)

• Examples of Riemannian Manifolds with non-negative sectional curvature