On Weyl-reducible conformal manifolds and lcK structures

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 x 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 Tq+1 . We show that in this case one necessarily has q = 1 or q = 2.