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Let $x$ : ${M}^{n-1}\rightarrow{\mathbb R}^{n}$ $(n\geq4)$ be an umbilical free hypersurface with non-zero principal curvatures. Then $x$ is associated with a Laguerre metric $\mathbf{g}$, a Laguerre tensor $\mathbf{L}$, a Laguerre form $\mathbf{C}$, and a Laguerre second fundamental form $\mathbf{B}$, which are invariants of $x$ under Laguerre transformation group. We denote the Laguerre scalar curvature by $R$ and the trace-free Laguerre tensor by $\tilde{\mathbf{L}}:={\mathbf{L}}-\frac{1}{n-1}tr({\mathbf{L}})\mathbf{g}$. In this paper, we prove a local classification result under the assumption of parallel Laguerre form and an inequality of the type $$\|\tilde{\mathbf{L}}\|\leq cR,$$ where $c=\frac{1}{(n-3)\sqrt{(n-2)(n-1)}}$ is appropriate real constant, depending on the dimension.
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참고문헌 (9)
참고문헌 신청
W. Blaschke / 1929 / Vorlesungen uber Differentialgeometrie / Springer-Verlag
T. E. Cecil / 1989 / Differential geometry and topology / Springer :1~48
Z. Guo / 2011 / Hypersurfaces with isotropic Blaschke tensor / J. Math. Soc. Japan 63(4):1155~1186
T. Li / 2005 / Laguerre geometry of surfaces in R3 / Acta Math. Sin. (Engl. Ser.) 21(6):1525~1534
T. Li / 2010 / Classification of hypersurfaces with parallel Laguerre second fundamental form in Rn / Differential Geom. Appl 28(2):148~157
T. E. Cecil / 1989 / Differential geometry and topology / Springer :1~48
Z. Guo / 2011 / Hypersurfaces with isotropic Blaschke tensor / J. Math. Soc. Japan 63(4):1155~1186
T. Li / 2005 / Laguerre geometry of surfaces in R3 / Acta Math. Sin. (Engl. Ser.) 21(6):1525~1534
T. Li / 2010 / Classification of hypersurfaces with parallel Laguerre second fundamental form in Rn / Differential Geom. Appl 28(2):148~157
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