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LAGUERRE CHARACTERIZATIONS OF HYPERSURFACES IN ℝn

Shu, Shichang;Li, Yanyan

  • Received : 2011.05.22
  • Published : 2013.11.30

Abstract

Let x : $M{\rightarrow}\mathbb{R}^n$ be an n - 1-dimensional hypersurface in $\mathbb{R}^n$, L be the Laguerre Blaschke tensor, B be the Laguerre second fundamental form and $D=L+{\lambda}B$ be the Laguerre para-Blaschke tensor of the immersion x, where ${\lambda}$ is a constant. The aim of this article is to study Laguerre Blaschke isoparametric hypersurfaces and Laguerre para-Blaschke isoparametric hypersurfaces in $\mathbb{R}^n$ with three distinct Laguerre principal curvatures one of which is simple. We obtain some classification results of such isoparametric hypersurfaces.

Keywords

Laguerre characterization;Laguerre form;Laguerre Blaschke tensor;Laguerre second fundamental form

References

  1. W. Blaschke, Vorlesungenuber Differential geometrie, Springer, Berlin, Heidelberg, New York, Vol. 3, 1929.
  2. G. H. Li, Mobius hypersurfaces in $S^{n+1}$ with three distinct principal curvatures, J. Geom. 80 (2004), no. 1-2, 154-165.
  3. T. Z. Li, Laguerre geometry of surfaces in $\mathbb{R}^3$, Acta Math. Sin. (Engl. Ser.) 21 (2005), no. 6, 1525-1534. https://doi.org/10.1007/s10114-005-0642-1
  4. T. Z. Li, H. Z. Li, and C. P. Wang, Classification of hypersurfaces with parallel Laguerre second fundamental form in $\mathbb{R}^n$, Differential Geom. Appl. 28 (2010), no. 2, 148-157. https://doi.org/10.1016/j.difgeo.2009.09.005
  5. T. Z. Li and C. P. Wang, Laguerre geometry of hypersurfaces in $\mathbb{R}^n$, Manuscripta Math. 122 (2007), no. 1, 73-95.
  6. E. Musso and L. Nicolodi, A variational problem for surfaces in Laguerre geometry, Trans. Amer. Math. Soc. 348 (1996), no. 11, 4321-4337. https://doi.org/10.1090/S0002-9947-96-01698-4
  7. E. Musso and L. Nicolodi, Laguerre geometry of surfaces with plane lines of curvature, Abh. Math. Sem. Univ. Hamburg 69 (1999), 123-138. https://doi.org/10.1007/BF02940867