Do Riemannian Submersions Preserve Positive Intermediate Ricci Curvature?

Abstract

Pro and the third author showed that there are Riemannian submersions $\pi: M \to B$ with $M$ a compact manifold with positive Ricci curvature, whose base $B$, has Ricci curvatures with both signs. Thus, Riemannian submersions need not preserve positive Ricci curvature. In this note we establish the degree to which this result extends into the setting of positive intermediate Ricci curvature. It is an immediate consequence of the Gray--O'Neill Horizontal curvature equation that if $\pi: M\to B$ is a Riemannian submersion whose base is $b$-dimensional and $\mathrm{Ric}_{k}(M) >0$ for any $k \in \{ 1,2,\cdots, b-1\}$, then $ \mathrm{Ric}_{k}(B)$ is also positive. Here we show that this observation is optimal in the following strong sense: For $k \geq \mathrm{dim}(B)$, let $\pi: (M,g_M) \to (B,g_B)$ be a Riemannian submersion from a complete Riemannian manifold with $\mathrm{Ric}_{k}(M) >0$. We show how to perturb $g_M$ in the $C^1$-topology to produce a Riemannian submersion $\pi: (M,\tilde{g}_M) \to (B,\tilde{g}_B)$ whose total space has $\mathrm{Ric}_{k} >0$, but whose base has Ricci curvature of both signs. In particular, this shows that Riemannian submersions that do not preserve positive Ricci curvature are dense in the $C^1$-topology among the complete metrics on $M$ with $\mathrm{Ric}>0$ for which a given submersion $\pi: M\to B$ is Riemannian.