Locally conformally product structures

Abstract

A locally conformally product (LCP) structure on compact manifold [Formula: see text] is a conformal structure [Formula: see text] together with a closed, non-exact and non-flat Weyl connection [Formula: see text] with reducible holonomy. Equivalently, an LCP structure on [Formula: see text] is defined by a reducible, non-flat, incomplete Riemannian metric [Formula: see text] on the universal cover [Formula: see text] of [Formula: see text], with respect to which the fundamental group [Formula: see text] acts by similarities. It was recently proved by Kourganoff that in this case [Formula: see text] is isometric to the Riemannian product of the flat space [Formula: see text] and an incomplete irreducible Riemannian manifold [Formula: see text]. In this paper, we show that for every LCP manifold [Formula: see text], there exists a metric [Formula: see text] such that the Lee form of [Formula: see text] with respect to [Formula: see text] vanishes on vectors tangent to the distribution on [Formula: see text] defined by the flat factor [Formula: see text], and use this fact in order to construct new LCP structures from a given one by taking products. We also establish links between LCP manifolds and number field theory, and use them in order to construct large classes of examples, containing all previously known examples of LCP manifolds constructed by Matveev–Nikolayevsky, Kourganoff and Oeljeklaus–Toma (OT-manifolds).