Locally conformally product structures on solvmanifolds (Q6618013)

A locally conformally product (LCP) structure on a compact connected manifold \(M\) is a non-flat Riemannian metric \(h\) with reducible holonomy on the universal cover \(\widetilde{M}\) such that \(\pi_1 (M)\) acts by homotheties with respect to \(h\), not of all which are isometries. In particular, \((\widetilde{M}, h)\) has to be incomplete, so the fundamental group of \(M\) is infinite.\N\NIt is worth noting that there is a strong similarity between LCP structures and locally conformally Kähler (LCK) structures, which is reflected in the terminology used. Indeed, an LCK structure on a compact connected manifold \(M\) can be defined as a Kähler metric \( h\) on the universal cover \(\widetilde{M}\) with respect to which \(\pi_1 (M)\) acts by homotheties, not all of which are isometries.\N\NConversely, every conformal manifold \((M, c)\) with a closed non-exact Weyl structure \(D\) with reducible but non-flat holonomy is LCP, the metric \(h\) on the universal cover being defined as the unique (up to a scalar factor) Riemannian metric on \(\widetilde{M}\) whose Levi-Civita connection is the lift of \(D\).\N\NThe authors study left-invariant locally conformally product structures on simply connected Lie groups and give their complete description in the solvable unimodular case. Based on previous classification results, they obtain the complete list of solvable unimodular Lie algebras up to dimension \(5\) which carry LCP structures, and study the existence of lattices in the corresponding simply connected Lie groups.