Nonpositively curved $4$-manifolds with zero Euler characteristic

TL;DR

This paper proves that closed nonpositively curved 4-manifolds with zero Euler characteristic exhibit Ricci curvature degeneration or foliations by flat 3-manifolds, with implications for their universal covers and conjectures on simplicial volume.

Contribution

It establishes new geometric constraints on nonpositively curved 4-manifolds with zero Euler characteristic, linking curvature degeneration, foliations, and Euclidean factors.

Findings

Ricci curvature degenerates somewhere on the manifold

Existence of foliations by flat 3-manifolds near each point

Universal cover has a Euclidean de Rham factor if the metric is analytic

Abstract

We show that for any closed nonpositively curved Riemannian 4-manifold $M$ with vanishing Euler characteristic, the Ricci curvature must degenerate somewhere. Moreover, for each point $p\in M$, either the Ricci tensor degenerates or else there is a foliation by totally geodesic flat 3-manifolds in a neighborhood of $p$. As a corollary, we show that if in addition the metric is analytic, then the universal cover of $M$ has a nontrivial Euclidean de Rham factor. Finally we discuss how this result creates an implication of conjectures on simplicial volume in dimension four.