伪黎曼空间型中子流形几何的若干问题
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摘要
本文研究了伪黎曼空间型中子流形几何的若干问题,给出了伪黎曼空间型中具有可对角化形算子的非退化超曲面的一些特征和分类结果,特别给出了r-极小超曲面的一些有趣特征.
一方面,利用活动标架法和L_r算子研究了伪黎曼空间型中满足一定条件的超曲面的特征问题:
1.伪黎曼空间型x:M_s~n→(?)(c)((?)R_(?)~(n+p)或R_(v+(?))~(n+p),p=1或2)中位置向量x满足方程L_rx=Rx+b的正常超曲面,其中L_r是相伴第r+1阶平均曲率H_(r+1)的二阶线性微分算子,r=0,…,n-1.R∈R~((n+1)×(n+1))或R~((n+2)×(n+2))是一个常矩阵,按照c=0或c≠0:b∈R_(?)~(n+p)或R_(v+1)~(n+p)是一个常向量.当超曲面满足以下条件之一时:(1)c=0;(2)c≠0.b=(?)且H_r是一个常数;(3)c≠0,b=(?)且R是自伴的,则仅有的满足上述条件的超曲面是r-极小的(即H_(r+1)≡0)或等参的.特别,我们局部分类了满足此条件的非r-极大类空超曲面,并推广了相关结果.
2.伪黎曼空间型(?)(c)中满足L_r(?)=λ(?)或L_rx=λ(?)等方程的非退化正常超曲面M_s~n,其中(?)是M_s~n在(?)(c)中的平均曲率向量场,λ∈R是常数.我们进一步给出了r-极小超曲面的一些新特征.
3.Lorentz空间型(?)(c)中满足方程φ=λψ,的完备类空超曲面,其中φ=和ψ=<(?),α>,α∈R_1~(n+1)或R_1~(n+2)或R_2~(n+2)是一个固定的非零向量,按照c=0,c=1或c=-1,λ是一个常数.如果超曲面具有常平均曲率,我们证明了满足此条件的超曲面M~n是一个全脐超曲面或一个双曲柱面.
此外,利用L_r算子讨论了Lorentz流形中具有常2阶平均曲率类空超曲面的稳定性.
另一方面,我们研究了Lorentz空间型(?)(c)中的满足一定曲率条件的类空超曲面,刻画了Lorentz空间型中具有两个互异主曲率的常r阶平均曲率类空超曲面的一些特征,得到了一些刚性结果.特别我们完全分类了反de Sitter空间中具有两个互异主曲率的常平均曲率的类空超曲面,解决了Cao和Wei提出的一个公开问题.
In this paper,we study some problems on geometry of submanifolds in pseudo-Riemannian space forms. We give some characteristics and classifications of proper hyper-surfaces in pseudo-Riemannian space forms. In particular, some interesting characteristics of r-minimal hypersurfaces are obtained.
On the one hand,using the method of moving frames and L_r operator, we investigate hypersurfaces x : M_s~n→(?) (c) immersed into a pseudo-Riemannian space form (?) (c) satisfying certain conditions on L_r operator:
1. Proper hypersurfaces in a pseudo-Riemannian space form (?) (c) ((?) R_v~(n+p) or R_(v+1)~(n+p)),whose position vector x satisfy L_rx = Rx + b,where L_r is the linearized operator of the (r + l)-th mean curvature H_(r+1) of hypersurfaces for a fixed r = 0,…,n-1, R∈R~((n+1)×(n+1)) or R~((n+2)×(n+2)) is a constant matrix,according to c = 0 or c≠0,respectively; and b∈R_v~(n+p) or R_(v+1)~(n+p) is a constant vector. If M_s~n satisfy one of the following properties:(1) c = 0, (2) c≠0,b = (?) and H_r is a constant, (3) c≠0, b = (?) and R is self-adjoint, then M_s~n are r-minimal (i.e. H_(r+1) = 0) or isoparametric. In particular, we locally classify such spacelike hypersurfaces which are not r-minimal.
2. Proper hypersurfaces M~n in a pseudo-Riemannian space form (?) (c) satisfying certain equation on the mean curvature vector (?) of M~n in pseudo-Riemannian space form (?) (c). We give some new characteristics of r-minimal hypersurfaces.
3. Complete spacelike hypersurface immersed into a Lorentzian space form (?) (c) satisfying equationφ=λψfor some real number A, whereφ= andψ= <(?),a>, for some fixed nonzero vector a∈R_1~(n+1),R_1~(n+2) or R_2~(n+2),according to c = 0, c = 1 or c = -1,respectively.We prove that if M~n has constant mean curvature, then M~n is either a totally umbilical hypersurface or a hyperbolic cylinder.
In adition. using L_r operator,we discuss stability of spacelike hypersurfaces with constant 2th mean curvature in a Lorentz manifolds.
On the other hand, we investigate spacelike hypersurfaces with constant rth mean curvature and two distinct principal curvatures in a Lorentz space forms (?) (c). We obtain some characteristics and rigity results of spacelike hypersurfaces in Lorentz space forms. In particular,we classify completely constant mean curvature spacelike hypersur- faces with two distinct principal curvatures in an anti-de Sitter space H_1~(n+1)(c) and solve a open problem presented by Cao and Wei.
一方面,利用活动标架法和L_r算子研究了伪黎曼空间型中满足一定条件的超曲面的特征问题:
1.伪黎曼空间型x:M_s~n→(?)(c)((?)R_(?)~(n+p)或R_(v+(?))~(n+p),p=1或2)中位置向量x满足方程L_rx=Rx+b的正常超曲面,其中L_r是相伴第r+1阶平均曲率H_(r+1)的二阶线性微分算子,r=0,…,n-1.R∈R~((n+1)×(n+1))或R~((n+2)×(n+2))是一个常矩阵,按照c=0或c≠0:b∈R_(?)~(n+p)或R_(v+1)~(n+p)是一个常向量.当超曲面满足以下条件之一时:(1)c=0;(2)c≠0.b=(?)且H_r是一个常数;(3)c≠0,b=(?)且R是自伴的,则仅有的满足上述条件的超曲面是r-极小的(即H_(r+1)≡0)或等参的.特别,我们局部分类了满足此条件的非r-极大类空超曲面,并推广了相关结果.
2.伪黎曼空间型(?)(c)中满足L_r(?)=λ(?)或L_rx=λ(?)等方程的非退化正常超曲面M_s~n,其中(?)是M_s~n在(?)(c)中的平均曲率向量场,λ∈R是常数.我们进一步给出了r-极小超曲面的一些新特征.
3.Lorentz空间型(?)(c)中满足方程φ=λψ,的完备类空超曲面,其中φ=
此外,利用L_r算子讨论了Lorentz流形中具有常2阶平均曲率类空超曲面的稳定性.
另一方面,我们研究了Lorentz空间型(?)(c)中的满足一定曲率条件的类空超曲面,刻画了Lorentz空间型中具有两个互异主曲率的常r阶平均曲率类空超曲面的一些特征,得到了一些刚性结果.特别我们完全分类了反de Sitter空间中具有两个互异主曲率的常平均曲率的类空超曲面,解决了Cao和Wei提出的一个公开问题.
In this paper,we study some problems on geometry of submanifolds in pseudo-Riemannian space forms. We give some characteristics and classifications of proper hyper-surfaces in pseudo-Riemannian space forms. In particular, some interesting characteristics of r-minimal hypersurfaces are obtained.
On the one hand,using the method of moving frames and L_r operator, we investigate hypersurfaces x : M_s~n→(?) (c) immersed into a pseudo-Riemannian space form (?) (c) satisfying certain conditions on L_r operator:
1. Proper hypersurfaces in a pseudo-Riemannian space form (?) (c) ((?) R_v~(n+p) or R_(v+1)~(n+p)),whose position vector x satisfy L_rx = Rx + b,where L_r is the linearized operator of the (r + l)-th mean curvature H_(r+1) of hypersurfaces for a fixed r = 0,…,n-1, R∈R~((n+1)×(n+1)) or R~((n+2)×(n+2)) is a constant matrix,according to c = 0 or c≠0,respectively; and b∈R_v~(n+p) or R_(v+1)~(n+p) is a constant vector. If M_s~n satisfy one of the following properties:(1) c = 0, (2) c≠0,b = (?) and H_r is a constant, (3) c≠0, b = (?) and R is self-adjoint, then M_s~n are r-minimal (i.e. H_(r+1) = 0) or isoparametric. In particular, we locally classify such spacelike hypersurfaces which are not r-minimal.
2. Proper hypersurfaces M~n in a pseudo-Riemannian space form (?) (c) satisfying certain equation on the mean curvature vector (?) of M~n in pseudo-Riemannian space form (?) (c). We give some new characteristics of r-minimal hypersurfaces.
3. Complete spacelike hypersurface immersed into a Lorentzian space form (?) (c) satisfying equationφ=λψfor some real number A, whereφ=
In adition. using L_r operator,we discuss stability of spacelike hypersurfaces with constant 2th mean curvature in a Lorentz manifolds.
On the other hand, we investigate spacelike hypersurfaces with constant rth mean curvature and two distinct principal curvatures in a Lorentz space forms (?) (c). We obtain some characteristics and rigity results of spacelike hypersurfaces in Lorentz space forms. In particular,we classify completely constant mean curvature spacelike hypersur- faces with two distinct principal curvatures in an anti-de Sitter space H_1~(n+1)(c) and solve a open problem presented by Cao and Wei.
引文
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[3] J. A. Aledo, L. J. Al(?)as and A. Romero, Integral formulas for compact spacelike hypersurfaces in de Sitter space: Applications to the case of constant higher order mean curvature, J. Geom.Phys., 31 (1999), 195-208.
[4] H. Alencar, M. do Carmo and A.G.Colares. Stable hyersurface with constant scalar curvature.Math. Z. 213 (1993), 117-131.
[5] H. Alencar, M. do Carmo and W. Santos, A gap theorem for hypersurfaces of the sphere with constant scalar curvature one,Comment. Math. Helv.77 (2002),549-562.
[6] L. J. Al(?)as, Characterization and Classification of Hypersurfaces in Pseudo-Riemannian Space Forms,Ph.D. Thesis. Universidad de Murcia, 1994.
[7] L. J. Al(?)as, S. C. De Almeida and A. Brasil Jr. Hpyersurfaces with constant mean curvature and two principal curvatures in S~(n+1).Anais da Academia Brasileira de Ciencias. 76 (2004),no. 3, 489-497.
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