The characteristic group of locally conformally product structures

TL;DR

This paper characterizes locally conformally product structures on compact manifolds with similarity structures, showing they are quotients of trivial bundles over simply-connected manifolds, and provides a classification in the case of simply connected characteristic groups.

Contribution

It describes LCP structures with simply connected characteristic groups as quotients of trivial bundles and proves their existence, advancing understanding of their geometric construction.

Findings

LCP structures with simply connected characteristic groups are quotients of trivial $R^p$-bundles.

Such quotients can always be endowed with an LCP structure.

The characteristic group plays a key role in classifying these 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 $\pi_1(M) \not\subset \mathrm{Isom}(\tilde M, h)$ 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 bundle 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.