局部对称空间中子流形的pinching问题
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摘要
子流形理论是微分几何中发展的比较成熟的分支学科.对子流形的第二基本形式模长平方s,数量曲率R,Ricci曲率R_(ii)及截面曲率R_(ijij)等内在量,加以某种限制,从而得到子流形的某些性质,叫做子流形的pinching问题.自从1968年J.Simons给出球面S~(n+p)(1)中极小子流形的积分公式后,几何学家对子流形的pinching问题研究的很多.
本文研究局部对称空间中子流形的pinching问题,全文共分为三个章节:
第一章节介绍局部对称空间中子流形的性质,从而为后面主要结果的证明作准备.
第二章节是关于局部对称空间中具有平行平均曲率向量子流形的pinching定理,设M是局部对称空间N~(n+p)中一个紧致子流形.我们应用Gauss方程,Ricci方程和外围空间的局部对称性质等方法,通过研究函数f(x)=(?)B(v.v)||~2得到一个pinching定理.当p≥2时,我们所得的—个定理改进了[1]中的相应结果.
第三章节研究局部对称空间中具有正Ricci曲率的完备极小子流形,得到了关于子流形Ricci曲率的一个pinching定理,该定理把Norio Ejiri的结论从外围空间为球空间推广到局部对称空间中.
The theory of submanifolds is a developed subject of differential geometry.If we give some restrician to the intrinsic quantities of submanifolds, such as second fundamental form, scalar curvature,Ricci curvature or sectional curvature,then we can get some new property of the submanifolds.The procedure is called pinching problem of submanifolds. In 1968,J.simons got the integral formulaof the minimal submanifolds of unit sphere S~(n+p)(1).After that time,many geometrician had got lots of results on the pinching problem of submanifolds.
We study a pinching theorem for submanifolds of locally symmetric space in this paper.This paper has three chapters.
In the first chapter, we give a brief introduction of the property of submanifoldsin locally symmetric space,which prepares for the proof of the following main results.
In the second chapter, we study a pinching theorem for submanifolds of locally symmetric space with parallel mean curvature .Let M be a compact submanifold of locally symmetric space N~(n+p),we apply Gauss equation,Ricci equation and the property of locally symmetry of the outer space ,through studying f(x) = (?) .then we get a pinching theorem.When p≥2,what we obtain improves the corresponding theorem of article[1].
In the third chapter, we study complete minimal submanifolds of locallysymmetric space, and we obtain a pinching theorem about the Ricci curvature of the minimal submanifolds, which generalizes the result of Norio Ejiri's from sphere space to locally symmetric space.
本文研究局部对称空间中子流形的pinching问题,全文共分为三个章节:
第一章节介绍局部对称空间中子流形的性质,从而为后面主要结果的证明作准备.
第二章节是关于局部对称空间中具有平行平均曲率向量子流形的pinching定理,设M是局部对称空间N~(n+p)中一个紧致子流形.我们应用Gauss方程,Ricci方程和外围空间的局部对称性质等方法,通过研究函数f(x)=(?)B(v.v)||~2得到一个pinching定理.当p≥2时,我们所得的—个定理改进了[1]中的相应结果.
第三章节研究局部对称空间中具有正Ricci曲率的完备极小子流形,得到了关于子流形Ricci曲率的一个pinching定理,该定理把Norio Ejiri的结论从外围空间为球空间推广到局部对称空间中.
The theory of submanifolds is a developed subject of differential geometry.If we give some restrician to the intrinsic quantities of submanifolds, such as second fundamental form, scalar curvature,Ricci curvature or sectional curvature,then we can get some new property of the submanifolds.The procedure is called pinching problem of submanifolds. In 1968,J.simons got the integral formulaof the minimal submanifolds of unit sphere S~(n+p)(1).After that time,many geometrician had got lots of results on the pinching problem of submanifolds.
We study a pinching theorem for submanifolds of locally symmetric space in this paper.This paper has three chapters.
In the first chapter, we give a brief introduction of the property of submanifoldsin locally symmetric space,which prepares for the proof of the following main results.
In the second chapter, we study a pinching theorem for submanifolds of locally symmetric space with parallel mean curvature .Let M be a compact submanifold of locally symmetric space N~(n+p),we apply Gauss equation,Ricci equation and the property of locally symmetry of the outer space ,through studying f(x) = (?) .then we get a pinching theorem.When p≥2,what we obtain improves the corresponding theorem of article[1].
In the third chapter, we study complete minimal submanifolds of locallysymmetric space, and we obtain a pinching theorem about the Ricci curvature of the minimal submanifolds, which generalizes the result of Norio Ejiri's from sphere space to locally symmetric space.
引文
[1] 陈卿:关于单位球面的子流形的一个pinching定理. 数学学报 39 (1996),57-63.
[2] 白正国,沈一兵,水乃翔等 黎曼几何初步 高等教育出版社 1992年4月.
[3] 李锦堂,林和子.关于局部对称空间中具有平行平均曲率向量子流形的pinching定理.厦门大学学报47 No.1(2008):16-19
[4] Norio EJIRI .Compact minimal submanifolds of a sphere with positive ricci curvature . J.Math.Soc.Jan 31(1979),251-256.
[5] 纪永强.子流形几何.科学出版社, 2004.
[6] Goldberg S I. Curvature and homology[M]. Academic press,London, 1962.
[7] Xu,H.W.On closed minimal submanifolds in pinched Riemannian manifolds, Trans.Amer.Math.Soc,347(1995).
[8] 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.
[9] 陈伟恒,李兴校.黎曼几何引论 上册 北京大学出版社 2002年12月
[2] 白正国,沈一兵,水乃翔等 黎曼几何初步 高等教育出版社 1992年4月.
[3] 李锦堂,林和子.关于局部对称空间中具有平行平均曲率向量子流形的pinching定理.厦门大学学报47 No.1(2008):16-19
[4] Norio EJIRI .Compact minimal submanifolds of a sphere with positive ricci curvature . J.Math.Soc.Jan 31(1979),251-256.
[5] 纪永强.子流形几何.科学出版社, 2004.
[6] Goldberg S I. Curvature and homology[M]. Academic press,London, 1962.
[7] Xu,H.W.On closed minimal submanifolds in pinched Riemannian manifolds, Trans.Amer.Math.Soc,347(1995).
[8] 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.
[9] 陈伟恒,李兴校.黎曼几何引论 上册 北京大学出版社 2002年12月