Focal sets and real hypersurfaces in complex projective space

Authors:
Thomas E. Cecil and Patrick J. Ryan

Journal:
Trans. Amer. Math. Soc. **269** (1982), 481-499

MSC:
Primary 53C40; Secondary 53C15

MathSciNet review:
637703

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Abstract: Let be a real submanifold of , and let denote the complex structure. We begin by finding a formula for the location of the focal points of in terms of its second fundamental form. This takes a particularly tractable form when is a complex submanifold or a real hypersurface on which is a principal vector for each unit normal to . The rank of the focal map onto a sheet of the focal set of is also computed in terms of the second fundamental form. In the case of a real hypersurface on which is principal with corresponding principal curvature , if the map onto a sheet of the focal set corresponding to has constant rank, then that sheet is a *complex* submanifold over which 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 . First, there are no totally umbilic real hypersurfaces in , but we show:

Theorem 3. *Let* *be a connected real hypersurface in* , , *with at most two distinct principal curvatures at each point. Then* *is an open subset of a geodesic hypersphere*. Secondly, we show that there are no Einstein real hypersurfaces in 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.

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DOI:
https://doi.org/10.1090/S0002-9947-1982-0637703-3

Keywords:
Focal sets,
real hypersurfaces in complex projective space,
umbilic hypersurfaces,
Einstein hypersurfaces

Article copyright:
© Copyright 1982
American Mathematical Society