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Hopf Hypersurfaces in pseudo-Riemannian complex and para-complex space forms
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In this paper, the authors studied real hypersurfaces in pseudo-Riemannian complex space forms and para-complex space forms, and proved that there are no umbilic hypersurface with parallel shape operator in such spaces.Abstract:
The study of real hypersurfaces in pseudo-Riemannian complex space forms and para-complex space forms, which are the pseudo-Riemannian generalizations of the complex space forms, is addressed. It is proved that there are no umbilic hypersurfaces, nor real hypersurfaces with parallel shape operator in such spaces. Denoting by $J$ be the complex or para-complex structure of a pseudo-complex or para-complex space form respectively, a non-degenerate hypersurface of such space with unit normal vector field $N$ is said to be \em Hopf \em if the tangent vector field $JN$ is a principal direction. It is proved that if a hypersurface is Hopf, then the corresponding principal curvature (the \em Hopf \em curvature) is constant. It is also observed that in some cases a Hopf hypersurface must be, locally, a tube over a complex (or para-complex) submanifold, thus generalizing previous results of Cecil, Ryan and Montiel.read more
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Journal ArticleDOI
Focal sets and real hypersurfaces in complex projective space
Thomas E. Cecil,Patrick J. Ryan +1 more
TL;DR: In this paper, the location of the focal points of a real submanifold is defined in terms of its second fundamental form, and the rank of a focal map onto a sheet of focal points corresponding to a principal curvature is computed using the Codazzi equation.
Book ChapterDOI
Real Hypersurfaces in Complex Space Forms
Thomas E. Cecil,Patrick J. Ryan +1 more
TL;DR: The study of real hypersurfaces in complex projective space CP n and complex hyperbolic space CH n began at approximately the same time as Munzner's work on isoparametric hypersurface in spheres as discussed by the authors.
Journal ArticleDOI
Real hypersurfaces of a complex hyperbolic space
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