Cauchy-Riemann submanifolds of locally conformal Kähler manifolds. II. (English) Zbl 0674.53057
[Part I, cf. Geom. Dedicata 28, No.2, 181-197 (1988; Zbl 0659.53041).]
A generic submanifold is a CR-submanifold \(M^ n\) such that dim \(D^{\perp}=co\dim M^ n\). In the paper the author proves that a locally conformal Kähler manifold \(M^{2m}\) does not admit proper generic submanifolds with a parallel f-structure and a nowhere vanishing 1-form induced by the Lee form of \(M^{2m}\). Hence \(M^ n\) must be totally real since dim \(D^{\perp}\geq 1\). We also remark the following result: any proper totally umbilical CR-submanifold \(M^ n\) of a locally conformal Kähler manifold \(M^{2m}\) is totally geodesic, provided the Lee field of \(M^{2m}\) is tangent to \(M^ n\).
A generic submanifold is a CR-submanifold \(M^ n\) such that dim \(D^{\perp}=co\dim M^ n\). In the paper the author proves that a locally conformal Kähler manifold \(M^{2m}\) does not admit proper generic submanifolds with a parallel f-structure and a nowhere vanishing 1-form induced by the Lee form of \(M^{2m}\). Hence \(M^ n\) must be totally real since dim \(D^{\perp}\geq 1\). We also remark the following result: any proper totally umbilical CR-submanifold \(M^ n\) of a locally conformal Kähler manifold \(M^{2m}\) is totally geodesic, provided the Lee field of \(M^{2m}\) is tangent to \(M^ n\).
Reviewer: A.Bejancu
MSC:
| 53C40 | Global submanifolds |
| 53B35 | Local differential geometry of Hermitian and Kählerian structures |
