Riemannian Submersions Need Not Preserve Positive Ricci Curvature

TL;DR

This paper explores how Riemannian submersions from positively Ricci curved manifolds can lead to base manifolds with negative Ricci curvature, revealing limitations in curvature preservation.

Contribution

It demonstrates that Riemannian submersions do not necessarily preserve positive Ricci curvature, contrasting with known results for sectional curvature.

Findings

Existence of submersions from positive Ricci to negatively curved neighborhoods

No submersions from positive Ricci to nonpositive Ricci manifolds

Positive Ricci curvature is not preserved under Riemannian submersions

Abstract

If $\pi :M\rightarrow B$ is a Riemannian Submersion and $M$ has positive sectional curvature, O'Neill's Horizontal Curvature Equation shows that $B$ must also have positive curvature. We show there are Riemannian submersions from compact manifolds with positive Ricci curvature to manifolds that have small neighborhoods of (arbitrarily) negative Ricci curvature, but that there are no Riemannian submersions from manifolds with positive Ricci curvature to manifolds with nonpositive Ricci curvature.