Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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LAGUERRE CHARACTERIZATION OF SOME HYPERSURFACES

  • Fang, Jianbo (School of Mathematics and Statistics Chuxiong Normal University) ;
  • Li, Fengjiang (Department of Mathematics Yunnan Normal University)
  • Received : 2015.05.22
  • Published : 2016.05.31
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Abstract

Let x : $^{Mn-1}{\rightarrow}{\mathbb{R}}^n$ ($n{\geq}4$) be an umbilical free hyper-surface with non-zero principal curvatures. Then x is associated with a Laguerre metric g, a Laguerre tensor L, a Laguerre form C, and a Laguerre second fundamental form B, which are invariants of x under Laguerre transformation group. We denote the Laguerre scalar curvature by R and the trace-free Laguerre tensor by ${\tilde{L}}:=L-{\frac{1}{n-1}}tr(L)g$. In this paper, we prove a local classification result under the assumption of parallel Laguerre form and an inequality of the type $${\parallel}{\tilde{L}}{\parallel}{\leq}cR$$ where $c={\frac{1}{(n-3){\sqrt{(n-2)(n-1)}}}$ is appropriate real constant, depending on the dimension.

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References

  1. W. Blaschke, Vorlesungen uber Differentialgeometrie, Berlin, Springer-Verlag, 1929.
  2. T. E. Cecil and S. S. Chern, Dupin submanifolds in Lie sphere geometry, Differential geometry and topology (Tianjin, 19867), 1-48, Lecture Notes in Math., 1369, Springer, Berlin, 1989.
  3. Z. Guo, J. Fang, and L. Lin, Hypersurfaces with isotropic Blaschke tensor, J. Math. Soc. Japan 63 (2011), no. 4, 1155-1186. https://doi.org/10.2969/jmsj/06341155
  4. T. Li, Laguerre geometry of surfaces in $R^{3}$, Acta Math. Sin. (Engl. Ser.) 21 (2005), no. 6, 1525-1534. https://doi.org/10.1007/s10114-005-0642-1
  5. T. Li, H. Li, and C. Wang, Classification of hypersurfaces with parallel Laguerre second fundamental form in $R^{n}$, Differential Geom. Appl. 28 (2010), no. 2, 148-157. https://doi.org/10.1016/j.difgeo.2009.09.005
  6. T. Li, H. Li, and C. Wang, Classification of hypersurfaces with constant Laguerre eigenvalues in $R^n$, Sci. China Math. 54 (2011), no. 6, 1129-1144. https://doi.org/10.1007/s11425-011-4170-4
  7. T. Li and C. Wang, Laguerre geometry of hypersurfaces in $\mathbb{R}^n$, Manuscripta Math. 122 (2007), no. 1, 73-95. https://doi.org/10.1007/s00229-006-0058-y
  8. H. Liu, C. Wang, and G. Zhao, Mobius isotropic submanifolds in $S^{n}$, Tohoku Math. J. (2) 53 (2001), no. 4, 553-569. https://doi.org/10.2748/tmj/1113247800
  9. C. Wang, Mobius geometry of submanifolds in $S^{n}$, Manuscripta Math. 96 (1998), no. 4, 517-534. https://doi.org/10.1007/s002290050080