On some real hypersurfaces of a complex projective space

Abstract

A principal circle bundle over a real hypersurface of a complex projective space CPn can be regarded as a hypersurface of an odddimensional sphere. From this standpoint we can establish a method to translate conditions imposed on a hypersurface of CPn into those imposed on a hypersurface of S2'+1. Some fundamental relations between the second fundamental tensor of a hypersurface of CPn and that of a hypersurface of S2n+1 are given. Introduction. As is well known a sphere S2n+1 of dimension 2n + 1 is a principal circle bundle over a complex projective space CPn and the Riemannian structure on CPn is given by the submersion ir: S2n+ 1 ~ CPn [4], [7]. This suggests that fundamental properties of a submersion would be applied to the study of real submanifolds of a complex projective space. In fact, H. B. Lawson [2] has made one step in this direction. His idea is to construct a principal circle bundle M2n over a real hypersurface M2n-1 of Cpn in such a way that M2n is a hypersurface of S2n + 1 and then to compare the length of the second fundamental tensors of M2n-1 and M2 n. Thus we can apply theorems on hypersurfaces of S2n+1. In this paper, using Lawson's method, we prove a theorem which characterizes some remarkable classes of real hypersurfaces of Cpn. First of all, in ?1, we state a lemma for a hypersurface of a Riemannian manifold of constant curvature for the later use. In ?2, we recall fundamental formulas of a submersion which are obtained in [4], [7] and those established between the second fundamental tensors of M and M. In ?3, we give some identities which are valid in a real hypersurface of CPn. After these preparations, we show, in ?4, a geometric meaning of the commutativity of the second fundamental tensor of M in Cp'n and a fundamental tensor of the submersion ir: M . M 1. Hypersurfaces of a Riemannian manifold of constant curvature. Let M be an (m + 1)-dimensional Riemannian manifold with a Riemannian metric G and i: M M be an isometric immersion of an m-dimensional differentiable Received by the editors March 25, 1974 and, in revised form, September 9, 1974. AMS (MOS) subject classifications (1970). Primary 53C40, 53C20.