Pseudo-Riemannian manifolds with nonzero parallel Weyl tensor which are not locally symmetric are known as ECS manifolds. Every ECS manifold carries a distinguished null parallel distribution \(\mathcal {D}\), the rank \(d\in \{1,2\}\) of which is referred to as the rank of the manifold itself. Under a natural genericity assumption on the Weyl tensor, we fully describe the universal coverings of compact rank-one ECS manifolds. We then show that any generic compact rank-one ECS manifold must be translational, in the sense that the holonomy group of the natural flat connection induced on \(\mathcal {D}\) is either trivial or isomorphic to \({\mathbb {Z}}_2\). We also prove that all four-dimensional rank-one ECS manifolds are noncompact, this time without having to assume genericity, as it is always the case in dimension four.

具有非局部对称的非零平行 Weyl 张量的伪黎曼流形称为 ECS 流形。每个 ECS 流形都带有一个独特的零并行分布\(\mathcal {D}\)，其秩\(d\in \{1,2\}\)被称为流形本身的秩。在 Weyl 张量的自然泛性假设下，我们充分描述了紧致一级 ECS 流形的通用覆盖。然后我们证明任何通用紧致一阶 ECS 流形都必须是平移的，在这个意义上，在\(\mathcal {D}\)上诱导的自然平坦连接的完整群要么是平凡的，要么同构于\({\mathbb {Z}}_2\)。我们还证明所有四维一阶 ECS 流形都是非紧的，这次不必假设通用性，因为在四维中总是如此。