Rn上主曲率非零的定向无脐超曲面x:M→Rn称为 Laguerre 等参超曲面,如果它的 Laguerre形式C=∑iCiwi=∑iρ-1(Ei(logρ)(r-ri)-Ei(r))wi为零,Laguerre 形状算子 S=ρ-1(S-rid的特征值为常数, 这里ρ2=∑i(r-ri)2,r=r1+r2+…+rn-1/n-1是平均曲率半径,S是x的形状算子, Ei是Laguerre度量g的单位正交标架, wi是对偶标架.本文给出Rn上具有三个互异Laguerre主曲率的Laguerre等参超曲面的分类.
Abstract
An umbilical free oriented hypersurface x:M→Rn with nonzero princi-pal curvatures is called Laguerre isoparametric hypersurface if its Laguerre form C=∑i Ciwi =∑iρ-1(Ei(logρ)(r-ri)-Ei(r))wi vanishes and Laguerre shape operator S=ρ-1(S-rid has constant eigenvalues. Here ρ2=∑i(r-ri)2,r=r1+r2+…+rn-1/n-1 is the mean curvature radius and S is the shape operator of x, Ei is local orthogonal basis for Laguerre metric g with dual basis wi. In this paper, we classify all Laguerre isoprametric hypersurfaces in Rn with three distinct Laguerre principal curvatures up to Laguerre transformation.
关键词
Laguerre 等参超曲面 /
Laguerre 度量 /
Laguerre第二基本形式 /
Laguerre 张量
{{custom_keyword}} /
Key words
Laguerre isoparametric hypersurface /
Laguerre metric /
Laguerre second fundamental form /
Laguerre tensor
{{custom_keyword}} /
{{custom_sec.title}}
{{custom_sec.title}}
{{custom_sec.content}}
参考文献
[1] Blaschke W., Vorlesungen über Differentialgeometrie, Vol.3, Springer-Verlag, Berlin, 1929.
[2] Cecil T., Jensen J. R., Dupin hypersurfaces with principal curvatures, Innvent. Math., 1998, 132: 121-178.
[3] Hu Z. J., Zhai S. J., Möbius isoparametric hypersurfaces in Sn+1 with three distinct principal curvatures, Pacific J. Math., 2011, 249: 343-370.
[4] Li T. Z., Li H., Wang C. P., Classification of hypersurfaces with constant Laguerre eigenvalues in Rn, Science China Math., 2011, 54: 1129-1144.
[5] Li T. Z., Li H., Wang C. P., Classification of hypersurfaces with parallel Laguerre second fundamental form in Rn, Differential Geometry and its Applications, 2010, 28: 148-157.
[6] Li T. Z., Wang C. P., Laguerre geometry of hypersurfaces in Rn, Maunscripta Math., 2007, 122: 73-95.
[7] Magid M. A., Lorentzian isoparametric hypersurfaces, Pacific J. Math., 1985, 118(1): 165-197.
[8] Musso E., Nicolodi L., A variational problem for surfaces in Laguerre geometry, Trans. Amer. Math., Soc., 1996, 348: 4321-4337.
[9] Musso E., Nicolodi L., Laguerre geometry of surfaces with plane lines of curvature, Abh. Math. Sem. Univ. Hamburg, 1999, 69: 123-138.
[10] Musso E., Nicolodi L., The Bianchi-Darboux transformation of L-isothermic surfaces, Intern. J. Math., 2000, 11(7): 911-924.
[11] Palmer B., Remarks on variation problem in Laguerre geometry, Rendicoonti di Mathematica, Serie VII 19 Toma, 1999: 281-293.
[12] Pinkall P., Dupin hypersurfaces, Math. Ann., 1985, 270: 427-440.
[13] Song Y. P., Laguerre isoparametric hypersurfaces in Rn with two distinct non-zero principal curvatures, Acta Math. Sinica, English Series, 2013, 29: 1-12.
{{custom_fnGroup.title_cn}}
脚注
{{custom_fn.content}}
基金
国家自然科学基金资助项目(11361004);江西省自然科学基金(20122BAB201014,20132BAB21107)及江西省科技厅(GJJ13659)资助项目
{{custom_fund}}