Semi-Kaehlerian submanifolds of an indefinite complex space form

Abstract

The purpose of this paper is to study several classes of semi-Kaehlerian submanifolds of an indefinite complex space form. Introduction. An indefinite Kaehlerian manifold of constant holomorphic sectional curvature is called an indefinite complex space form. Montiel and Romero [9] investigated indefinite complex Einstein hypersurfaces of an indefinite complex space forms and showed that totally geodesic indefinite complex hypersurfaces Pί(c'), C?, H(—c) and an indefinite complex quadric Qf are those examples. Ikawa, Romero and one of present authors [6] have also shown that by using an indefinite Segre imbedding there exists a product of complex hyperbolic spaces which becomes an example of space-like Einstein-Kaehlerian submanifolds of an indefinite complex hyperbolic space. Recently, concerning with the study of Calabi's classification [5] for Kaehlerian imbeddings of complex space forms into complex space forms, Romero [18] and Umehara [21] have independently found that there exists a strongly full holomorphic isometric immersion of indefinite complex space forms into indefinite complex space forms. From this point of view the purpose of this paper is to study several classes of complete semi-Kaehlerian submanifolds of an indefinite complex space form M#nc' In the first section, the brief summary of indefinite complex submanifolds of an indefinite Kaehlerian manifold are recalled. The examples of space-like complex Einstein submanifolds of indefinite complex space forms are given in § 2. § 3 is devoted to the study of the spacelike complex submanifolds with constant scalar curvature of M%(c'). In particular, by estimating the scalar curvature and by using Nishikawa's theorem [12], we shall characterize space-like Einstein submanifolds in the case of c'<0. Received December 8, 1987