局部对称空间中的超曲面和局部对称Lorentz空间中的类空超曲面
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摘要
自黎曼几何诞生以来,黎曼流形的研究一直成为黎曼几何研究的核心内容。对外围空间具有良好对称性的黎曼流形中子流形的研究特别是对球空间中子流形的研究已经获得了非常丰富的结果。当把外围空间推广到其它情形如局部对称空间或Lorentz空间时,对此类空间中的黎曼流形的研究,许多学者已经获得了不少具有重要价值的研究成果。
本文利用外微分法、活动标架法、积分法以及局部和整体相结合的方法研究了局部对称空间中的超曲面和局部对称Lorentz空间中的类空超曲面及De Sitter空间中的线性Weingarten类空超曲面的各种内在量之间所蕴含的几何性质,并获得了内在量满足某种关系时的内蕴刚性定理,具体地讲,本文的主要研究成果有以下几个方面。
1.研究了局部对称空间中具有常数量曲率的超曲面,利用丘成桐教授的自伴随算子给出了此超曲面的第二基本形式满足一定刚性条件时全脐的内蕴刚性定理,从而推广了李海中教授的一个结论。
2.首先给出了局部对称Lorentz空间的定义,利用丘成桐教授和Omori教授的广义极大值原理以及自伴随算子分别研究了此空间中具有常平均曲率的完备类空超曲面和常数量曲率的紧致类空超曲面,得出了两个内蕴刚性定理,把欧阳崇祯、李镇琦和刘西民等人的主要结果推广到局部对称Lorentz空间。
3.首先引入de Sitter空间中线性Weingarten类空超曲面的概念并利用丘成桐和陈省身的自伴随算子口研究了此线性Weingarten类空超曲面,并得到了两个重要的内蕴刚性定理。其推广了Zheng[19]和Li[20]关于de Sitter空间中具有常平均曲率和常数量曲率类空超曲面的两个重要结果。
The study on the Riemannian manifold is an important field of the Riemannian geometry since Riemannian geometry had been born. When the ambient space has good symmetric, the submanifold of Riemannian manifold, especially submanifold in a sphere had been studied and obtained many rigidity theorems. While the ambient space is extended to locally symmetric space or locally symmetric Lorentz space, many researchers had obtained many important results.
In this paper, by making use of the exterior differentiation, moving frames method and integral methods, we study the geometric properties of hypersurfaces or space-like hyper-surfaces in a locally symmetric space or a locally symmetric Lorentz space and obtain some rigidity theorems under its intrinsic invariant satisfying some intrinsic conditions, in detail, the main results in this paper may be stated as follows:
1. The hypersurfaces with constant scalar curvature in locally symmetric space are studied by making use of the self operator of Yau S T and some rigidity theorems are obtained if the squared norm of the second fundamental form satisfying some pinching condition, the result of Hai Zhong Li is generalized.
2. The definition of the locally symmetric Lorentz space is introduced, complete hypersurfaces with constant mean curvature and the compact space-like hypersurfaces with constant scalar curvature are studied by making use of the exterior differentiation, moving frames method. Two rigidity theorems are obtained and the main results of Chong Zhen OU-YANG, Zhen Qi Li and Xin Min Liu are generalized.
3. First introduce the new define of the linear weingarten space-like hypersurfaces in de Sitter space and by making use of the self-adjiont differential operator of S.Y.Cheng and S.T.Yau study the linear weingarten space-like hypersurfaces we get two important rigidity theorems, which generalized the two main results of zheng[19] and Li[20] on the space-like hypersurfaces with constant mean curvature and constant scalar curvature in de sitter space.
本文利用外微分法、活动标架法、积分法以及局部和整体相结合的方法研究了局部对称空间中的超曲面和局部对称Lorentz空间中的类空超曲面及De Sitter空间中的线性Weingarten类空超曲面的各种内在量之间所蕴含的几何性质,并获得了内在量满足某种关系时的内蕴刚性定理,具体地讲,本文的主要研究成果有以下几个方面。
1.研究了局部对称空间中具有常数量曲率的超曲面,利用丘成桐教授的自伴随算子给出了此超曲面的第二基本形式满足一定刚性条件时全脐的内蕴刚性定理,从而推广了李海中教授的一个结论。
2.首先给出了局部对称Lorentz空间的定义,利用丘成桐教授和Omori教授的广义极大值原理以及自伴随算子分别研究了此空间中具有常平均曲率的完备类空超曲面和常数量曲率的紧致类空超曲面,得出了两个内蕴刚性定理,把欧阳崇祯、李镇琦和刘西民等人的主要结果推广到局部对称Lorentz空间。
3.首先引入de Sitter空间中线性Weingarten类空超曲面的概念并利用丘成桐和陈省身的自伴随算子口研究了此线性Weingarten类空超曲面,并得到了两个重要的内蕴刚性定理。其推广了Zheng[19]和Li[20]关于de Sitter空间中具有常平均曲率和常数量曲率类空超曲面的两个重要结果。
The study on the Riemannian manifold is an important field of the Riemannian geometry since Riemannian geometry had been born. When the ambient space has good symmetric, the submanifold of Riemannian manifold, especially submanifold in a sphere had been studied and obtained many rigidity theorems. While the ambient space is extended to locally symmetric space or locally symmetric Lorentz space, many researchers had obtained many important results.
In this paper, by making use of the exterior differentiation, moving frames method and integral methods, we study the geometric properties of hypersurfaces or space-like hyper-surfaces in a locally symmetric space or a locally symmetric Lorentz space and obtain some rigidity theorems under its intrinsic invariant satisfying some intrinsic conditions, in detail, the main results in this paper may be stated as follows:
1. The hypersurfaces with constant scalar curvature in locally symmetric space are studied by making use of the self operator of Yau S T and some rigidity theorems are obtained if the squared norm of the second fundamental form satisfying some pinching condition, the result of Hai Zhong Li is generalized.
2. The definition of the locally symmetric Lorentz space is introduced, complete hypersurfaces with constant mean curvature and the compact space-like hypersurfaces with constant scalar curvature are studied by making use of the exterior differentiation, moving frames method. Two rigidity theorems are obtained and the main results of Chong Zhen OU-YANG, Zhen Qi Li and Xin Min Liu are generalized.
3. First introduce the new define of the linear weingarten space-like hypersurfaces in de Sitter space and by making use of the self-adjiont differential operator of S.Y.Cheng and S.T.Yau study the linear weingarten space-like hypersurfaces we get two important rigidity theorems, which generalized the two main results of zheng[19] and Li[20] on the space-like hypersurfaces with constant mean curvature and constant scalar curvature in de sitter space.
引文
[1]H Alencar and M do Carmo. Hypersurfaces with constant mean curvature in sphere[J]. Proceedings of the American Mathematics Society,1994,120,1223-1229
[2]S S Chern, M. Do Carmo and S kobayashi. Minimal Submanifolds of a sphere with second fundamental form of constant length[C]. Functional Analysis and Related Fields, Springer-Verlag, Berlin,1970,59-75
[3]S Y Cheng and S T Yau. Hypersurfaces with constant scalar curvature[J]. Annals of Mathematics.1977,225,195-204
[4]C Shu S and S Y Liu. Complete Hypersurfaces with Constant Mean Curvature in Locally Symmetric Manifold [J]. Advances in Mathematics(Chinese),2004,33,563-569
[5]Hlineva S and Belchev E. On the minimal hypersurfaces of a locally symmetric manifold[J]. Lecture Notes in Mathematics,1990,1481,1-4
[6]水乃翔,吴国强.局部对称黎曼流形中的极小子流形[J].数学年刊,1995,16A(6)：687-691
[7]陈卿。 局部对称空间的极小超曲面[J].科学通报,1993,38,1057-1059
[8]马金生.关于局部对称空间中具有常数量曲率的超曲面[J].安徽大学学报,2002,26,12-17
[9]H Li. Hypersurfaces with constant scalar curvature in space forms[J]. Annals of Mathematics,1996,35,665-672
[10]宋卫东。关于局部对称空间中的极小子流形[J].数学年刊,1998,19A(6)：693-698
[11]Akutagawa K.:On space-like hypersurfaces with constant mean curvature in the de Sitter space[J]. Mathematics, Z,1987,196,13-19
[12]Chong-zhen OU YANG and Li zhen qi. Complete space-like hypersurfaces with constant mean curvature in de Sitter spaces[J]. Journal of Mathematics,2000,15(2):45-49
[13]Ramanathan J.:Complete space-like hypersurfaces of constant mean curvature in the de Sitter spaces[J]. Zndiana University journal of Mathematics,1987,36,349-359
[14]Xi min Liu. Complete space-like hypersurfaces with constant scalar curvature[J]. Manuscripta Math,2001,105(3):367-377.
[15]N Abe, N Koike and S Yamaguuchi. Rongruence theorems for proper smai-Riemannian hypersurfaces in a real space form[J]. Yokohama journal Mathematics,1987,35, 123-136
[16]Yau S T. Harmonic functions on complete Riemannian manifolds [J]. Commun pure and Applied Mathematics,1975,28:201-241
[17]Omori H. Isometric immersion of Riemmaian manifolds [J]. Journal of the Japan Mathematical Society,1967,19:205-214
[18]ALDIR Brasil J, A Gervasio Colares, Oscar Palmas. A gap theorem for complete constant scalar curvature hypersurfaces in the de Sitter space [J]. Journal of Geometry and physics,2001,37:237-250
[19]Y Zheng. Space-like hypersurfaces with constant scalar curvature in the de Sitter spaces[J], Differential Goemetry and its Application,1996,6:51-54
[20]H Li. Global rigidity theorems of hypersurfaces [J], Ark. Mathematics,1997(35), no.2, 327-351
[21]Cheng Q M and Ishikawa S. Spaces-like Hypersurfaces with constant scalar curvature[J]. Manuscripta Mathematics,1998,95,499-505
[22]Liu X. Spaces-like hypersurfaces of constant scalar curvature in the de Sitter spaces[J]. Atti Sem. Mat. Fis. Univ. Modena,2000,48,89-106
[23]Montiel S. An integral inequality for compact space-like hypersurfaces in the de Sitter spaces and applications to the case of constant mean curvature[J]. Indiana University Mathematics Journal,1998,37,909-917
[24]Zheng Y. On space-like hypersurfaces in the de Sitter spaces[J], Annals of Global Analysis and Geometry,1995,13,317-321
[25]X Liu. Space-like submanifolds with constant scalar curvature[J], Differential Geometry. Complete Riemanne. Academica Science Paris,2001,729-734
[26]M Okumura. Hypersurfaces and a pinching problem on the sencond fundamental tensor[J], American Journal Mathematics,1974,96,207-213
[27]Hai Zhong Li, Young Jin Suh and Guo Xin Wei, linear weigarten hypersurfaces in a unit sphere[J], Bulln Korean Mathematcs Society,2009,46, No. O,p.1
[2]S S Chern, M. Do Carmo and S kobayashi. Minimal Submanifolds of a sphere with second fundamental form of constant length[C]. Functional Analysis and Related Fields, Springer-Verlag, Berlin,1970,59-75
[3]S Y Cheng and S T Yau. Hypersurfaces with constant scalar curvature[J]. Annals of Mathematics.1977,225,195-204
[4]C Shu S and S Y Liu. Complete Hypersurfaces with Constant Mean Curvature in Locally Symmetric Manifold [J]. Advances in Mathematics(Chinese),2004,33,563-569
[5]Hlineva S and Belchev E. On the minimal hypersurfaces of a locally symmetric manifold[J]. Lecture Notes in Mathematics,1990,1481,1-4
[6]水乃翔,吴国强.局部对称黎曼流形中的极小子流形[J].数学年刊,1995,16A(6)：687-691
[7]陈卿。 局部对称空间的极小超曲面[J].科学通报,1993,38,1057-1059
[8]马金生.关于局部对称空间中具有常数量曲率的超曲面[J].安徽大学学报,2002,26,12-17
[9]H Li. Hypersurfaces with constant scalar curvature in space forms[J]. Annals of Mathematics,1996,35,665-672
[10]宋卫东。关于局部对称空间中的极小子流形[J].数学年刊,1998,19A(6)：693-698
[11]Akutagawa K.:On space-like hypersurfaces with constant mean curvature in the de Sitter space[J]. Mathematics, Z,1987,196,13-19
[12]Chong-zhen OU YANG and Li zhen qi. Complete space-like hypersurfaces with constant mean curvature in de Sitter spaces[J]. Journal of Mathematics,2000,15(2):45-49
[13]Ramanathan J.:Complete space-like hypersurfaces of constant mean curvature in the de Sitter spaces[J]. Zndiana University journal of Mathematics,1987,36,349-359
[14]Xi min Liu. Complete space-like hypersurfaces with constant scalar curvature[J]. Manuscripta Math,2001,105(3):367-377.
[15]N Abe, N Koike and S Yamaguuchi. Rongruence theorems for proper smai-Riemannian hypersurfaces in a real space form[J]. Yokohama journal Mathematics,1987,35, 123-136
[16]Yau S T. Harmonic functions on complete Riemannian manifolds [J]. Commun pure and Applied Mathematics,1975,28:201-241
[17]Omori H. Isometric immersion of Riemmaian manifolds [J]. Journal of the Japan Mathematical Society,1967,19:205-214
[18]ALDIR Brasil J, A Gervasio Colares, Oscar Palmas. A gap theorem for complete constant scalar curvature hypersurfaces in the de Sitter space [J]. Journal of Geometry and physics,2001,37:237-250
[19]Y Zheng. Space-like hypersurfaces with constant scalar curvature in the de Sitter spaces[J], Differential Goemetry and its Application,1996,6:51-54
[20]H Li. Global rigidity theorems of hypersurfaces [J], Ark. Mathematics,1997(35), no.2, 327-351
[21]Cheng Q M and Ishikawa S. Spaces-like Hypersurfaces with constant scalar curvature[J]. Manuscripta Mathematics,1998,95,499-505
[22]Liu X. Spaces-like hypersurfaces of constant scalar curvature in the de Sitter spaces[J]. Atti Sem. Mat. Fis. Univ. Modena,2000,48,89-106
[23]Montiel S. An integral inequality for compact space-like hypersurfaces in the de Sitter spaces and applications to the case of constant mean curvature[J]. Indiana University Mathematics Journal,1998,37,909-917
[24]Zheng Y. On space-like hypersurfaces in the de Sitter spaces[J], Annals of Global Analysis and Geometry,1995,13,317-321
[25]X Liu. Space-like submanifolds with constant scalar curvature[J], Differential Geometry. Complete Riemanne. Academica Science Paris,2001,729-734
[26]M Okumura. Hypersurfaces and a pinching problem on the sencond fundamental tensor[J], American Journal Mathematics,1974,96,207-213
[27]Hai Zhong Li, Young Jin Suh and Guo Xin Wei, linear weigarten hypersurfaces in a unit sphere[J], Bulln Korean Mathematcs Society,2009,46, No. O,p.1