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

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.

Torus actions on manifolds with positive intermediate Ricci curvature

“…Then the metric g * on 𝑆 3 × 𝑆 3 is left-invariant, it is invariant under the diagonal action of 𝑆 3 by right multiplication, and Ric 2 (𝑆 3 × 𝑆 3 , g * ) > 0. For more information about this construction, including a generalization to products of compact semisimple Lie groups, see Theorem E in [12]. It follows that symrank(𝑆 3 × 𝑆 3 , g * ) = 3, which is maximal for closed 6dimensional manifolds with Ric 2 > 0 by Theorem 1.1.…”

“…Then the metric $$g_*$$ on $$S^3\times S^3$$ is left‐invariant, it is invariant under the diagonal action of $$S^3$$ by right multiplication, and $$\operatorname{Ric}_2(S^3\times S^3,g_*)>0$$. For more information about this construction, including a generalization to products of compact semisimple Lie groups, see Theorem E in [12]. It follows that $$\operatorname{symrank}(S^3\times S^3,g_*) = 3$$, which is maximal for closed $$\hskip.001pt 6$$‐dimensional manifolds with $$\operatorname{Ric}_2>0$$ by Theorem 1.1.…”

“…See Section 2 for basic examples of manifolds with $$\operatorname{Ric}_k>0$$. For the classification of compact symmetric spaces by the minimal value of $$k$$ for which each has $$\operatorname{Ric}_k>0$$, see [2] or [12]. We note that the only compact irreducible symmetric spaces that have $$\operatorname{Ric}_2>0$$ are those of rank $$\hskip.001pt 1$$, that is, those that have positive sectional curvature.…”

“…We note that the only compact irreducible symmetric spaces that have $$\operatorname{Ric}_2>0$$ are those of rank $$\hskip.001pt 1$$, that is, those that have positive sectional curvature. For more constructions of $$\operatorname{Ric}_k>0$$, see Theorems E, F, and G in [12].…”

On the dynamics of positively curved metrics on SU(3)/T 2 under the homogeneous Ricci flow

“…Indeed, part of the idea to this note was conceived looking to the examples approached in [12]. These were built in [10] and provide metrics of intermediate positive Ricci curvature (in the sense of Definition 3.2) on some generalized Wallach spaces. In [12], the authors show that these conditions are not preserved under the homogeneous Ricci flow.…”