On the Inoue-Bombieri construction

TL;DR

This paper investigates compact quotients of product manifolds involving Euclidean and Riemannian components, establishing equivalences with LCP manifolds, rigidity results for symmetric spaces, and classifying cases with negatively curved Hadamard manifolds.

Contribution

It links Inoue-Bombieri constructions to LCP manifolds, proves a rigidity theorem for symmetric spaces, and classifies quotients when the manifold has negative curvature.

Findings

Equivalent to LCP manifolds

Bieberbach-type rigidity for symmetric spaces

Classification for negatively curved Hadamard manifolds

Abstract

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.