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Conformally flat hypersurfaces with constant Gauss-Kronecker curvature

Published online by Cambridge University Press:  17 April 2009

Filip Defever
Affiliation:
Departement Wiskunde, KU Leuven, Celestijnenlaan 200 B, 3001 Heverlee (Leuven), Belgium, e-mail: [email protected]
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Abstract

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We consider 3-dimensional conformally flat hypersurfaces of E4 with constant Gauss-Kronecker curvature. We prove that those with three different principal curvatures must necessarily have zero Gauss-Kronecker curvature.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2000

References

REFERENCES

[1]Brito, F., Liu, H.L., Simon, U. and Wang, C.P., ‘Hypersurfaces in space forms with some constant curvature functions’, Geom. Topol. Submanifolds 9 (1999), 4863.CrossRefGoogle Scholar
[2]Chen, B.-Y., Geometry of submanifolds (Marcel Dekker, New York, 1973).Google Scholar
[3]Defever, F., ‘Conformally flat hypersurfaces of E4 with constant mean curvature’, (preprint).Google Scholar
[4]Do Carmo, M., Dajczer, M. and Mercuri, F., ‘Compact conformally flat hypersurfaces’, Trans. Amer. Math. Soc 288 (1985), 189203.CrossRefGoogle Scholar
[5]Hertrich-Jeromin:, U., Über konform flache Hyperflächen in vierdimensionalen Raumformen, PhD thesis (Tuebingen University, Berlin, 1994).Google Scholar
[6]Kohlmann, P., Uniqueness results for hypersurfaces in space forms, (Habilitation thesis) (University Dortmund, Dortmund, 1995).Google Scholar
[7]Kulkarni, R.S., ‘Conformally flat manifolds’, Proc. Nat. Acad. Sci. U.S.A 69 (1972), 26752676.CrossRefGoogle ScholarPubMed
[8]Lancaster, G.M., ‘Canonical metrics for certain conformally Euclidean spaces of dimension 3 and codimension 1’, Duke Math. J 40 (1973), 18.CrossRefGoogle Scholar
[9]Pinkall, U., ‘Compact conformally flat hypersurfaces’, in Conformal geometry, (Kulkarni, R.S. and Pinkall, U., Editors) (Max Planck Institut für Mathematik, Bonn, 1988), pp. 217236.CrossRefGoogle Scholar