On the normal bundle of a complex submanifold of a locally conformal Kaehler manifold
Let \(M\) be a complex submanifold of a locally conformal Kähler manifold \(\widetilde{M}\). The author derives a sufficient condition in order that the normal bundle of \(M\) admits no nontrivial parallel section. Next, suppose that the Lee form of \(\widetilde{M}\) is parallel, that is, \(\widetilde{M}\) is a generalized Hopf manifold, and that \(M\) is tangent to the Lee field of \(\widetilde{M}\). When \(M\) has flat normal connection, the author derives an equation relating the Ricci tensors of \(M\), \(\widetilde{M}\) and gets as a consequence that \(M\) is quasi-Einstein if \(\widetilde{M}\) is quasi-Einstein. The author also obtains sufficient conditions in order that the first Chern class of the normal bundle vanishes or the normal connection is flat.
scientific article; zbMATH DE number 872715
scientific article