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

• Published in: Mathematische annalen Vol. 386; no. 3-4; pp. 1979 - 2014
• Main Authors: Domínguez-Vázquez, Miguel, González-Álvaro, David, Mouillé, Lawrence
• Format: Journal Article
• Language: English
• Published: 01.08.2023
• ISSN: 0025-5831; 1432-1807
• DOI: 10.1007/s00208-022-02420-w

Abstract Positive $$k\mathrm{th}$$ k th -intermediate Ricci curvature on a Riemannian n -manifold, to be denoted by $${{\,\mathrm{Ric}\,}}_k>0$$ Ric k > 0 , is a condition that interpolates between positive sectional and positive Ricci curvature (when $$k =1$$ k = 1 and $$k=n-1$$ k = n - 1 respectively). In this work, we produce many examples of manifolds of $${{\,\mathrm{Ric}\,}}_k>0$$ 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 $$n\ge 7$$ n ≥ 7 congruent to $$3\,{{\,\mathrm{mod}\,}}4$$ 3 mod 4 supports infinitely many closed simply connected manifolds of pairwise distinct homotopy type, all of which admit homogeneous metrics of $${{\,\mathrm{Ric}\,}}_k>0$$ Ric k > 0 for some $$k<n/2$$ k < n / 2 . We also prove that each dimension $$n\ge 4$$ n ≥ 4 congruent to 0 or $$1\,{{\,\mathrm{mod}\,}}4$$ 1 mod 4 supports closed manifolds which carry metrics of $${{\,\mathrm{Ric}\,}}_k>0$$ Ric k > 0 with $$k\le n/2$$ k ≤ n / 2 , but do not admit metrics of positive sectional curvature.