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 $\Gamma$ of $\mathrm{Sim}(\mathbb{R}^q) \times \mathrm{Isom}(N)$. 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 $\Gamma$ when $N$ is a Hadamard manifold with strictly negative curvature.