Do Riemannian Submersions Preserve Positive Intermediate Ricci Curvature?
Hasan M. El-Hasan, Russell Phelan, Frederick Wilhelm
TL;DR
This work investigates whether Riemannian submersions preserve positive intermediate Ricci curvature. It proves that for k ≤ b−1, positive Ric_k on the total space M transfers to positive Ric_k on the base B via the Gray–O’Neill equation, and for k ≥ b, one can perturb metrics in the C^1-topology to obtain a base with Ric_k taking both signs while keeping Ric_k^M>0; moreover, such non-preserving submersions are dense among complete Ric>0 metrics making the submersion Riemannian. The main construction uses a two-level deformation: a conformal change on the base and a horizontal/vertical warping on M guided by carefully controlled C^1-small functions; the Reiser–Wraith lemma is employed to guarantee the persistence of Ric_k^M>0 under perturbation. Collectively, the paper shows that positivity of intermediate Ricci curvature is not robust under submersion in higher k, and non-preserving instances are abundant in the C^1 topology.
Abstract
Pro and the third author showed that there are Riemannian submersions $π: M \to B$ 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 $π: M\to B$ 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 $π: (M,g_M) \to (B,g_B)$ 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 $π: (M,\tilde{g}_M) \to (B,\tilde{g}_B)$ 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 $π: M\to B$ is Riemannian.