On the normal bundle of a complex submanifold of a locally conformal Kaehler manifold
type: Item
instance of: scholarly article
title: On the normal bundle of a complex submanifold of a locally conformal Kaehler manifold
zbMATH Open document ID: 0859.53010
DOI: 10.1007/BF01223033
author: Sorin Dragomir
published in: Journal of Geometry
publication date: 1996-06-13T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 53B25
Mathematics Subject Classification ID: 53C15
zbMATH DE Number: 872715
zbMATH Keywords: Ricci tensor
zbMATH Keywords: flat normal connection
zbMATH Keywords: complex submanifold
zbMATH Keywords: locally conformal Kähler manifold
zbMATH Keywords: generalized Hopf manifold
zbMATH Keywords: Lee field
zbMATH Keywords: quasi-Einstein
zbMATH Keywords: Chern class of the normal bundle
reviewed by: Jürgen Berndt
MaRDI profile type: Publication
cites work: Some Fundamental Formulas for Hermitian Manifolds with Non-Vanishing Torsion
cites work: Q5678153
cites work: Cauchy-Riemann submanifolds of locally conformal Kähler manifolds
cites work: Q3828682
cites work: Q4039129
cites work: A new upper bound for the complex Grothendieck constant
cites work: Q3473754
cites work: Generalized Hopf manifolds, locally conformal Kaehler structures and real hypersurfaces
cites work: Holomorphic bisectional curvature
cites work: On compact locally conformal Kaehler manifolds with non-negative sectional curvature
cites work: Q5731354
cites work: Differential geometry of complex hypersurfaces
cites work: Generalized Hopf manifolds
cites work: Q3892907
cites work: Some Curvature Properties of Locally Conformal Kahler Manifolds
cites work: Q4185444
Wikidata QID: Q126107054
description: scientific article; zbMATH DE number 872715