CR-submanifolds of a complex space form

Abstract

Publisher Summary This chapter presents some fundamental formulas for submanifolds of a Kaehlerian manifold, and in particular for those of a complex space form, and discusses the CR-submanifolds and generic submanifolds. The chapter discusses various theorems related to CR-submanifolds. Theorem 1 is fundamental in the study of CR-submanifolds. The f -structures are studied in which a CR-submanifold and its normal bundle admit. Theorem 2 characterizes generic submanifolds with parallel f -structure of a complex space form. An integral formula of Simons' type is derived and applying it to prove Theorems 3, 4 and 5. It is clear that every real hypersurface of a Kaehlerian manifold is automatically generic submanifold. A CR-submanifold is called a “proper CR-submanifold” if it is neither a holomorphic submanifold nor an anti-invariant submanifold.