Classes of Hypersurfaces with Vanishing Laplace Invariants
Bull. Korean Math. Soc. 2012 Vol. 49, No. 4, 685-692
https://doi.org/10.4134/BKMS.2012.49.4.685
Published online July 1, 2012
Carlos M. C. Riveros and Armando M. V. Corro
Universidade de Bras\'ilia, Universidade Federal de Goi\^as
Abstract : Consider a hypersurface $M^n$ in $ \mathbb{R}^{n+1}$ with $n$ distinct principal curvatures, parametrized by lines of curvature with vanishing Laplace invariants. (1) If the lines of curvature are planar, then there are no such hypersurfaces for $n\geq 4$, and for $n=3$, they are, up to M\"obius transformations, Dupin hypersurfaces with constant M\"obius curvature. (2) If the principal curvatures are given by a sum of functions of separated variables, there are no such hypersurfaces for $n\geq 4$, and for $n=3$, they are, up to M\"obius transformations, Dupin hypersurfaces with constant M\"obius curvature.
Keywords : lines of curvature, Laplace invariants, Dupin hypersurfaces
MSC numbers : Primary 53A07; Secondary 35N10
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