  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.
