Mixed foliate CR-submanifolds of indefinite complex space-forms
Let M be a CR-submanifold of a Kähler manifold \(\tilde M\) and D and \(D^{\perp}\) be the distributions carried by M [see the reviewer, Geometry of CR-submanifolds (Reidel 1986; Zbl 0605.53001)]. Then M is called mixed foliate if D is integrable and the second fundamental form h of M satisfies \(h(X,Y)=0\), for any X and Y from D and \(D^{\perp}\) respectively. In the present paper a mixed foliate proper CR-submanifold M of an indefinite complex space-form \(\tilde M(c)\) is considered and it is proved that \(c=0\), provided the restrictions of the metric to D and \(D^{\perp}\) are definite and indefinite, respectively. More results are obtained with respect to CR-submanifolds which have flat normal connection and parallelizable totally real distribution.
scientific article; zbMATH DE number 4063986