The characteristic group of locally conformally product structures

Abstract

A compact manifold M together with a Riemannian metric h on its universal cover \(\tilde{M}\) for which \(\pi _1(M)\) acts by similarities is called a similarity structure. In the case where \(\pi _1(M) \not \subset \textrm{Isom}(\tilde{M}, h)\) and \((\tilde{M}, h)\) is reducible but not flat, this is a Locally Conformally Product (LCP) structure. The so-called characteristic group of these manifolds, which is a connected abelian Lie group, is the key to understand how they are built. We focus in this paper on the case where this group is simply connected, and give a description of the corresponding LCP structures. It appears that they are quotients of trivial \(\mathbb {R}^p\)-principal bundles over simply-connected manifolds by certain discrete subgroups of automorphisms. We prove that, conversely, it is always possible to endow such quotients with an LCP structure.