Introduction

The study of real hypersurfaces in complex projective space ℂPn and complex hyperbolic space ℂHn has been an active field over the past decade. Although these ambient spaces might be regarded as the simplest after the spaces of constant curvature, they impose significant restrictions on the geometry of their hypersurfaces. For instance, they do not admit umbilic hypersurfaces and their geodesic spheres do not have constant curvature. They also do not admit Einstein hypersurfaces. M. Okumura [1978] remarked that there was a poverty of vocabulary for describing the differential geometric properties of the hypersurfaces that can arise. That situation has since been improved.

One can regard ℂPn as a projection from S 2 n + 1 with fibre S1. H. B. Lawson [1970] was the first to exploit this idea to study a hypersurface in ℂPn by lifting it to an S1-invariant hypersurface of the sphere. He identified certain hypersurfaces called equators of ℂPn which are minimal and lift to Clifford minimal hypersurfaces of the sphere. Subsequently, other investigators explored properties that lifted to familiar properties of hypersurfaces in S2 n + 1.