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CLASSES OF HYPERSURFACES WITH VANISHING LAPLACE INVARIANTS

  • Riveros, Carlos M.C. (Departamento de Matematica Universidade de Brasilia) ;
  • Corro, Armando M.V. (Instituto de Matematica e Estatiıstica Universidade Federal de Goias)
  • Received : 2011.03.18
  • Published : 2012.07.31
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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$\ddot{o}$bius transformations Dupin hypersurfaces with constant M$\ddot{o}$bius 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$\ddot{o}$bius transformations, Dupin hypersurfaces with constant M$\ddot{o}$bius curvature.

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References

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