On the normal bundle of a complex submanifold of a locally conformal Kaehler manifold

Abstract

We investigate the existence of parallel sections in the normal bundle of a complex submanifold of a locally conformal Kaehler manifold with positive holomorphic bisectional curvature. Also, if∼M is a quasi-Einstein generalized Hopf manifold then we show that any complex submanifoldM with a flat normal connection of∼M is quasi-Einstein, too, provided thatM is tangent to the Lee field of∼M. As an application of our results we study the geometry of the second fundamental form of a complex submanifold in the locally conformal Kaehler sphereQ m (of a complex Hopf manifoldS2m+1 ×S1).