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 G of a simply connected Riemannian Lie group (G, g) with Lie algebra g can be characterized in terms of a closed 1-form ^* and a non-zero subspace u g satisfying some algebraic conditions. We show that these conditions are equivalent to the fact that g 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 previous results holding for solvmanifolds. In addition, we construct explicit examples of compact LCP manifolds which are not solvmanifolds.