On Weyl-reducible conformal manifolds and lcK structures

Abstract

A recent result of M. Kourganoff states that if D is a closed, reducible, non-flat Weyl connection on a compact conformal manifold M , then the universal cover of M , endowed with the metric whose Levi-Civita covariant derivative is the pull-back of D, is isometric to R q × N for some irreducible, incomplete Riemannian manifold N . Moreover, he characterized the case where the dimension of N is 2 by showing that M is then a mapping torus of some Anosov diffeomorphism of the torus T q+1 . We show that in this case one necessarily has q = 1 or q = 2.