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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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 $ 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.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1982-0637703-3
PII: S 0002-9947(1982)0637703-3
Keywords: Focal sets, real hypersurfaces in complex projective space, umbilic hypersurfaces, Einstein hypersurfaces
Article copyright: © Copyright 1982 American Mathematical Society