On the Inoue-Bombieri construction

We study compact quotients of a Riemannian product ${\mathbb{R}}^q\times (N,g_N)$, where $(N,g_N)$ is a complete Riemannian manifold, by discrete subgroups $\mathrm{\Gamma }$ of $\mathrm{S}\mathrm{i}\mathrm{m}({\mathbb{R}}^q)\times \mathrm{I}\mathrm{s}\mathrm{o}\mathrm{m}(N)$. When $N$ is a symmetric space of non-compact type, this construction generalizes the well-known Inoue–Bombieri surfaces. We show that this setting is actually equivalent to that of the so-called LCP manifolds, and we establish a Bieberbach-type rigidity result in the case where $N$ is symmetric. In addition, we provide a classification of the manifolds $N$ and the groups $\mathrm{\Gamma }$ when $N$ is a Hadamard manifold with strictly negative curvature.