Hopf hypersurfaces in pseudo-Riemannian complex and para-complex space forms |
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Affiliation: | 1. Université Libre de Bruxelles, Géométrie Différentielle, C.P. 213 1050, Brussels, Belgium;2. Faculty of Mathematics and Engineering Sciences, Hellenic Military Academy, Vari, Attiki, Greece |
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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 Hopf 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 Hopf 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. |
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Keywords: | Real hypersurfaces Hopf hypersurfaces Tubes Pseudo-Riemannian geometry |
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