An umbilical free oriented hypersurfacex：M→Rnwith non-zero principal curvatures is called a Laguerre isoparametric hypersurface if its Laguerre form C=i Ciωi=iρ1（Ei（logρ）（r ri）Ei（r））ωi vanishes and Lague...An umbilical free oriented hypersurfacex：M→Rnwith non-zero principal curvatures is called a Laguerre isoparametric hypersurface if its Laguerre form C=i Ciωi=iρ1（Ei（logρ）（r ri）Ei（r））ωi vanishes and Laguerre shape operator S=ρ1（S 1 rid）has constant eigenvalues.Hereρ=i（r ri）2,r=r1＋r2＋···＋rn 1n 1is the mean curvature radius andSis the shape operator ofx.{Ei}is a local basis for Laguerre metric g=ρ2III with dual basis{ωi}and III is the third fundamental form ofx.In this paper,we classify all Laguerre isoparametric hypersurfaces in Rn（n〉3）with two distinct non-zero principal curvatures up to Laguerre transformations.展开更多
A hypersurface x: M→S^(n+1) without umbilic point is called a Mbius isoparametric hypersurface if its Mbius form Φ=-ρ^(-2)∑_i(ei(H)+∑_j(h_(ij)-Hδ_(ij))e_j(logρ))θ_i vanishes and its Mbius shape operator S=ρ^(...A hypersurface x: M→S^(n+1) without umbilic point is called a Mbius isoparametric hypersurface if its Mbius form Φ=-ρ^(-2)∑_i(ei(H)+∑_j(h_(ij)-Hδ_(ij))e_j(logρ))θ_i vanishes and its Mbius shape operator S=ρ^(-1)(S-Hid) has constant eigenvalues. Here {e_i} is a local orthonormal basis for I=dx·dx with dual basis {θ_i}, II=∑_(ij)h_(ij)θ_iθ_J is the second fundamental form, H=1/n∑_i h_(ij), ρ~2=n/(n-1)(‖II‖~2-nH^2) and S is the shape operator of x. It is clear that any conformal image of a (Euclidean) isoparametric hypersurface in S^(n+1) is a Mbius isoparametric hypersurface, but the converse is not true. In this paper we classify all Mbius isoparametric hypersurfaces in S^(n+1) with two distinct principal curvatures up to Mbius transformations. By using a theorem of Thorbergsson [1] we also show that the number of distinct principal curvatures of a compact Mbius isoparametric hypersurface embedded in S^(n+1) can take only the values 2, 3, 4, 6.展开更多
基金Supported by National Natural Science Foundation of China(Grant No.10826062)Natural Science Foundation of Fujian Province of China(Grant No.2012J01020)the Fundamental Research Funds for the Central Universities(Grant No.2011121040)
文摘An umbilical free oriented hypersurfacex：M→Rnwith non-zero principal curvatures is called a Laguerre isoparametric hypersurface if its Laguerre form C=i Ciωi=iρ1（Ei（logρ）（r ri）Ei（r））ωi vanishes and Laguerre shape operator S=ρ1（S 1 rid）has constant eigenvalues.Hereρ=i（r ri）2,r=r1＋r2＋···＋rn 1n 1is the mean curvature radius andSis the shape operator ofx.{Ei}is a local basis for Laguerre metric g=ρ2III with dual basis{ωi}and III is the third fundamental form ofx.In this paper,we classify all Laguerre isoparametric hypersurfaces in Rn（n〉3）with two distinct non-zero principal curvatures up to Laguerre transformations.
基金Partially supported by NSFCPartially supported by TU Berlin, DFG, SRF, SEM+2 种基金Partially supported by Qiushi Award. 973 Project, RFDPthe Jiechu GrantPartially supported by DFG, NSFC and Qiushi Award
文摘A hypersurface x: M→S^(n+1) without umbilic point is called a Mbius isoparametric hypersurface if its Mbius form Φ=-ρ^(-2)∑_i(ei(H)+∑_j(h_(ij)-Hδ_(ij))e_j(logρ))θ_i vanishes and its Mbius shape operator S=ρ^(-1)(S-Hid) has constant eigenvalues. Here {e_i} is a local orthonormal basis for I=dx·dx with dual basis {θ_i}, II=∑_(ij)h_(ij)θ_iθ_J is the second fundamental form, H=1/n∑_i h_(ij), ρ~2=n/(n-1)(‖II‖~2-nH^2) and S is the shape operator of x. It is clear that any conformal image of a (Euclidean) isoparametric hypersurface in S^(n+1) is a Mbius isoparametric hypersurface, but the converse is not true. In this paper we classify all Mbius isoparametric hypersurfaces in S^(n+1) with two distinct principal curvatures up to Mbius transformations. By using a theorem of Thorbergsson [1] we also show that the number of distinct principal curvatures of a compact Mbius isoparametric hypersurface embedded in S^(n+1) can take only the values 2, 3, 4, 6.
