The structure of locally conformally product Lie algebras

TL;DR

This paper characterizes the algebraic structure of locally conformally product Lie algebras on compact manifolds, extending previous results and providing explicit examples of such manifolds beyond solvmanifolds.

Contribution

It establishes an algebraic characterization of LCP structures as semidirect products involving non-unimodular Lie algebras and constructs explicit non-solvmanifold examples.

Findings

LCP structures correspond to semidirect product Lie algebras with specific properties.

The algebraic conditions for LCP structures are equivalent to certain conformal representations.

Explicit examples of compact LCP manifolds that are not solvmanifolds are provided.

Abstract

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 $\Gamma\backslash G$ of a simply connected Riemannian Lie group $(G,g)$ with Lie algebra $\mathfrak{g}$ can be characterized in terms of a closed 1-form $\theta\in\mathfrak{g}^*$ and a non-zero subspace $\mathfrak{u}\subset \mathfrak{g}$ satisfying some algebraic conditions. We show that these conditions are equivalent to the fact that $\mathfrak{g}$ 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.