Focal sets and real hypersurfaces in complex projective space

Abstract

Let $M$ be a real submanifold of $C{P^m}$, and let $J$ denote the complex structure. We begin by finding a formula for the location of the focal points of $M$ in terms of its second fundamental form. This takes a particularly tractable form when $M$ is a complex submanifold or a real hypersurface on which $J\xi$ is a principal vector for each unit normal $\xi$ to $M$. The rank of the focal map onto a sheet of the focal set of $M$ is also computed in terms of the second fundamental form. In the case of a real hypersurface on which $J\xi$ is principal with corresponding principal curvature $\mu$, if the map onto a sheet of the focal set corresponding to $\mu$ has constant rank, then that sheet is a complex submanifold over which $M$ is a tube of constant radius (Theorem 1). The other sheets of the focal set of such a hypersurface are given a real manifold structure in Theorem 2. These results are then employed as major tools in obtaining two classifications of real hypersurfaces in $C{P^m}$. First, there are no totally umbilic real hypersurfaces in $C{P^m}$, but we show: Theorem 3. Let $M$ be a connected real hypersurface in $C{P^m}$, $m \geqslant 3$, with at most two distinct principal curvatures at each point. Then $M$ is an open subset of a geodesic hypersphere. Secondly, we show that there are no Einstein real hypersurfaces in $C{P^m}$ and characterize the geodesic hyperspheres and two other classes of hypersurfaces in terms of a slightly less stringent requirement on the Ricci tensor in Theorem 4.