On Locally Conformal Kaehler Submersions

Abstract

We study locally conformal Kaehler submersions, i.e., almost Hermitian submersions whose total manifolds are locally conformal Kaehler. We prove that the vertical distribution of a locally conformal Kaehler submersion is totally geodesic iff the Lee vector field of total manifold is vertical. We also obtain the O'Neill tensors $\tilde{\mathcal{A}}$ and $\tilde{\mathcal{T}}$ with respect to the Weyl connection of a locally conformal Kaehler submersion. Then, we proved that the horizontal distribution of such a submersion is integrable iff $\tilde{\mathcal{A}} \equiv 0$. Finally, we get Chen-Ricci inequalities for locally conformal Kaehler space form submersions and Hopf space form submersions.