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arXiv · 2309.15766

Nonpositively curved $4$-manifolds with zero Euler characteristic

Chris Connell·Yuping Ruan·Shi Wang
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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.

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Chris Connell, Yuping Ruan, Shi Wang. 2023-09-27. Nonpositively curved $4$-manifolds with zero Euler characteristic. https://arxiv.org/abs/2309.15766

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