Real hypersurfaces in nonflat complex space forms are irreducible

Abstract

The study of real hypersurfaces in complex projective space and complex hyperbolic space has been an active field over the past three decades. 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 totally umbilical hypersurfaces and Einstein hypersurfaces. On the other hand, several important classes of real hypersurfaces in complex projective space have been constructed and investigated by many geometers. For instance, H.B. Lawson investigated real hypersurfaces of constructed by Clifford minimal hypersurfaces of +1 via Hopf fibration. R. Takagi [9] gave the list of homogeneous real hypersurfaces of . Many geometers then study the geometry from the list of Takagi and obtained various interesting geometric characterizations of homogeneous real hypersurfaces in . Another important class of real hypersurfaces in which contains the list of R. Takagi is the class of Hopf hypersurfaces. Such hypersurfaces are real hypersurfaces whose structure vector ξ is a principal curvature vector, where is the complex structure and ξ is the unit normal vector field. Examples and geometric characterizations of Hopf hypersurfaces have also been obtained by various geometers. It is known that in , is a homogeneous real hypersurface if and only if is a Hopf hypersurface with constant principal curvatures [6, 9]. The study of real hypersurfaces in complex hyperbolic space has followed developments in , often with similar results, but sometimes with differences (see [1, 7, 8] for more details). It is well-known that real projective space and real hyperbolic space admit ample hypersurfaces which are the Riemannian products of some Riemannian manifolds. It is also well-known that 3 admits a complex hypersurface which is the Riemannian