The structure of locally conformally product Lie algebras

A locally conformally product (LCP) structure on a compact conformal manifold is a closed non-exact Weyl connection (i.e. a linear connection which is locally but not globally the Levi-Civita connection of Riemannian metrics in the conformal class), with reducible holonomy. A left-invariant LCP structure on a compact quotient [Formula: see text] of a simply connected Riemannian Lie group [Formula: see text] with Lie algebra [Formula: see text] can be characterized in terms of a closed one-form [Formula: see text] and a non-zero subspace [Formula: see text][Formula: see text][Formula: see text] satisfying some algebraic conditions. We show that these conditions are equivalent to the fact that [Formula: see text] is isomorphic to a semidirect product of a non-unimodular Lie algebra acting on an abelian one by a conformal representation. This extends to the general case results from Andrada et al. [1] holding for solvmanifolds. In addition, we construct explicit examples of compact LCP manifolds which are not solvmanifolds.