Hypersurfaces with constant inner curvature of the second fundamental form,and the non-rigidity of the sphere |
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Authors: | Markus Becker Wolfgang Kühnel |
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Institution: | 1. Fachbereich Mathematik, Universit?t Duisburg, D-47048, Duisburg, Germany 2. Mathematisches Institut B, Universit?t Stuttgart, D-70550, Stuttgart, Germany
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Abstract: | We classify the hypersurfaces of revolution in euclidean space whose second fundamental form defines an abstract pseudo-Riemannian
metric of constant sectional curvature. In particular we find such piecewise analytic hypersurfaces of class C
2
where the second fundamental form defines a complete space of constant positive, zero, or negative curvature. Among them
there are closed convex hypersurfaces distinct from spheres, in contrast to a theorem of R. Schneider (Proc. AMS 35, 230–233,
(1972)) saying that such a hypersurface of class C
4
has to be a round sphere. In particular, the sphere is not II-rigid in the class of all convex C
2
-hypersurfaces.
Received 11 October 1994; in final form 26 April 1995 |
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Keywords: | |
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