LAGUERRE CHARACTERIZATION OF SOME HYPERSURFACES

Title & Authors
LAGUERRE CHARACTERIZATION OF SOME HYPERSURFACES
Fang, Jianbo; Li, Fengjiang;

Abstract
Let x : $\small{^{Mn-1}{\rightarrow}{\mathbb{R}}^n}$ ($\small{n{\geq}4}$) 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 ${\tilde{L}}: Keywords Laguerre geometry;hypersurfaces; Language English Cited by References 1. W. Blaschke, Vorlesungen uber Differentialgeometrie, Berlin, Springer-Verlag, 1929. 2. T. E. Cecil and S. S. Chern, Dupin submanifolds in Lie sphere geometry, Differential geometry and topology (Tianjin, 19867), 1-48, Lecture Notes in Math., 1369, Springer, Berlin, 1989. 3. Z. Guo, J. Fang, and L. Lin, Hypersurfaces with isotropic Blaschke tensor, J. Math. Soc. Japan 63 (2011), no. 4, 1155-1186. 4. T. Li, Laguerre geometry of surfaces in$R^{3}$, Acta Math. Sin. (Engl. Ser.) 21 (2005), no. 6, 1525-1534. 5. T. Li, H. Li, and C. Wang, Classification of hypersurfaces with parallel Laguerre second fundamental form in$R^{n}$, Differential Geom. Appl. 28 (2010), no. 2, 148-157. 6. T. Li, H. Li, and C. Wang, Classification of hypersurfaces with constant Laguerre eigenvalues in$R^n$, Sci. China Math. 54 (2011), no. 6, 1129-1144. 7. T. Li and C. Wang, Laguerre geometry of hypersurfaces in$\mathbb{R}^n$, Manuscripta Math. 122 (2007), no. 1, 73-95. 8. H. Liu, C. Wang, and G. Zhao, Mobius isotropic submanifolds in$S^{n}$, Tohoku Math. J. (2) 53 (2001), no. 4, 553-569. 9. C. Wang, Mobius geometry of submanifolds in$S^{n}\$, Manuscripta Math. 96 (1998), no. 4, 517-534.