On decomposable LCP structures (Q6661356)

Locally conformally product (LCP) manifolds are in many respects similar to locally conformally Kähler (LCK) manifolds. They can be defined either as compact quotients of simply connected (non-flat) Riemannian manifolds with reducible holonomy by discrete subgroups of homotheties not containing only isometries, or as compact conformal manifolds \((M, c)\) carrying a closed non-exact Weyl connection \(\nabla\) with reducible (non-zero) holonomy.\N\NAccording to a fundamental result of \textit{M. Kourganoff} [Math. Ann. 373, No. 3--4, 1075--1101 (2019; Zbl 1415.53020)], the tangent bundle of an LCP manifold \((M, c, \nabla)\) splits into two \(\nabla\)-parallel distributions, one of which is flat. A Riemannian metric \(g\) on \(M\) in the conformal class \(c\) is called adapted if the Lee form of \(\nabla\) with respect to \(g\) vanishes on the flat distribution. If \((M, c, \nabla)\) is an LCP structure, and \(g \in c\) is adapted, then for every compact Riemannian manifold \((K, g_K )\), the conformal manifold \((M \times K,[g + g_K ])\) carries an adapted LCP structure as well.\N\NThe LCP structures obtained in this way are called reducible, and it is obvious that the study of LCP structures can be reduced to understanding the irreducible ones. However, it might happen that an irreducible LCP manifold is weakly reducible, in the sense that it is obtained from a reducible LCP manifold by changing the action of the fundamental group on the universal cover.\N\NBecause of this phenomenon, the authors introduce the slightly more general notion of decomposable LCP structure which is, by definition, an LCP structure containing a Riemannian metric with reducible holonomy in the conformal class \(c\). A decomposable LCP structure \((M, c, \nabla)\) is locally reducible. The authors characterize those decomposable LCP manifolds which are defined on quotients of Riemannian Lie groups by co-compact lattices.