差异张量平行和半平行的仿射超曲面

本文用活动标架法证明了：若 Ｍｎ（ｎ≥２）是 ｎ＋１维仿射空间 Ａｎ＋１中非退化的仿射超曲面,（１）若■Ｋ＝０（即差异张量平行）,则Ｍ是仿射球,且Ｊ＝０和Ｇ是一个Ｅｉｎｓｔｅｉｎ度量,这里Ｊ是Ｍ的 Ｐｉｃｋ不变量,Ｇ是Ｂｌａｓｃｈｋｅ度量;（２）Ｒ·Ｋ＝０（即差异张量半平行）当且仅当Ｓ＝０（Ｍ为虚仿射球）,或者Ｋ＝０（Ｍ为非退化的二次超曲面）,这里 Ｒ为诱导仿射联络 ■的黎曼曲率算子．     

De Sitter空间中紧致类空超曲面的积分公式及其Goddard猜想的应用

本文对ｄｅＳｉｔｅｒ空间Ｓｎ＋１１（１）中紧致类空超曲面建立Ｍｉｎｋｏｗｓｋｉ型公式，并给出其到ｒ－次（ｒ＝１，２，…，ｎ－１）平均曲率为常数超曲面的应用．当ｒ＝１时，我们得到Ｍｏｎｔｉｅｌ的结果．     

全脐子流形的一个特征

１９７４年Ｍ．Ｏｋｕｍｕｒａ证明［１］：设Ｍｎ（ｎ≥３）为ｎ＋１维单位球面Ｓｎ＋１的紧致常平均曲率超曲面，若其第二基本形式的模长平方则Ｍｎ为全脐超曲面，其中Ｈ为平均曲率．本文目的是把限制常数改为，而且对于余维数大于１的情况也有类似的结果．     

CP~n中的弱等焦Hopf超曲面

本文研究了ＣＰｎ中的弱等焦Ｈｏｐｆ超曲面的分类，弱等焦超曲面首先是由滕楚莲与Ｔｈｏｒｂｅｒｇｓｓｏｎ在［１３］中引入的，设 Ｍ是 ＣＰｎ中的弱等焦 Ｈｏｐｆ超曲面，则 Ｍ的不同焦半径的个数是 ２，４或６；并且当 ｇ＝ ２或 ４时， Ｍ在 ＣＰｎ中是等参的；当 ｇ＝ ６时，ｎ＝ ３，Ｍ＝π－１（Ｍ）与 Ｓ７中的等参超曲面不Ｌｉｅ等价，此处π：Ｓ７→ＣＰ３是Ｈｏｐｆ纤维化．     

关于类空W超曲面的一个存在性定理

利用超曲面的旋转对称性，将ＰＤＥ的求解转化为ＯＤＥ的求解，确定了Ｄｅ Ｓｉｔｔｅｒ空间中的一类旋转型的Ｗｅｉｎｇａｒｔｅｎ超曲面。即获得：给定Ｒ＾ｎ－１内开集（０，∞）＾ｎ－１上一个Ｃ＾１函数ｋｎ＝ｆ（ｋ１，…，ｋｎ－１）（ｎ≥２），一定存在Ｄｅ Ｓｉｔｔｅｒ空间Ｓ１＾ｎ＋１内的ｎ维类空旋转超曲面Ｍ，使得Ｍ的ｎ个主曲率ｋ１，…，ｋｎ恰有上述函数关系。     

De Sitter空间中具有常均曲率的类空超曲面（Ⅱ）

设Ｍ＾ｎ是Ｄｅ Ｓｉｔｔｅｒ空间Ｓ１＾ｎ＋１中具有常数平均曲率且第二基本形式长度的平方为常数的完备类空超曲面，若Ｓ≤２（ｎ－１）＾１／２，则Ｍ＾ｎ是等参超曲面。     

平行三次形式的仿射超曲面

设Ａｎ 1是ｎ 1维仿射空间 ,Ｄ表示Ａｎ 1上的平坦联络 ,Ｍ是ｎ维光滑流形 ,ｘ:Ｍ→Ａｎ 1是一个非退化的仿射浸入 .对于Ｍ上的横截向量场ξ ,存在唯一的选择 ξ =1ｎ△ｘ(称为仿射法向量场 ) ,使得上述浸入是一个Ｂｌａｓｃｈｋｅ浸入 (见 [2 ]) .设 是此浸入由Ｄ在Ｍ上诱导的仿射联络 ,我们有 :ＤＸＹ= ＸＹ ｈ(Ｘ ,Ｙ) ξ　　　ＤＸξ=-ＳＸ这里Ｘ ,Ｙ ,Ｚ是Ｍ上的切向量场 ,ｈ是对称的双线性形式 ,由它可以定义Ｍ上的伪黎曼度量Ｇ(Ｇ =|Ｈ| - 1ｎ 2 ｈ ,Ｈ =ｄｅｔ(ｈｉｊ) ) ,称为Ｂｌａｓｃｈｋｅ度量 ,Ｓ称为Ｍ的形态算子 .若…     

具有共变对称张量场的仿射超曲面

用活动标架法从不同方面证明并推广了文[2]的结果：（1）若Mn（n≥2）是n＋1维仿射空间An 1中非退化的仿射超曲面，则2S共变对称（或R·S=0），当且仅当M是仿射球；（2）若Mn（n≥2）是An 1中非退化的仿射起曲面，则K共变对称当且仅当M是仿射球；若K和K都共变对称，则M是仿球，且J=0和仿射度量G是Einstein度量．     

保持高斯映射的仿射形变

给定Ｒｉｅｍａｎｎ流形到欧氏空间的仿射浸入ｆ：　Ｍｎ→ＲＮ　　，我们建立存在另一个与ｆ有相同高斯映射的仿射浸入ｆ：Ｍｎ→ＲＮ　　的条件，进一步利用这个条件，解答了仿射浸入的高斯映射将其确定到何种程度的问题．     

Lorentz空间中常平均曲率类空超曲面

本文证明了ｎ＋１维Ｌｏｒｅｎｔｚ空Ｌｎ＋１中以Ｓｎ－１（ｒ）为边界的紧致常平均曲率类空超曲面只有 Ｂｎ（ｒ）和超伪球面盖．对于 Ｒｎ＋１中的紧致常平均曲率超曲面，当高斯映照像落在一个半球面内时，得到相应的唯一性结果．     

Affine Locally Symmetric Surfaces in R~4

The nondegenerate affine locally symmetric surfaces in R^4 with the transversal bundle defined by Nomizu and Vrancken have been studied and a complete classification of the locally symmetric surfaces with flat normal bundle has been given.     

Four-Dimensional Simply Connected Symplectic Symmetric Spaces

A symplectic is a symmetric space endowed with a symplectic structure which is invariant by the symmetries. We give here a classification of four-dimensional symplectic which are simply connected. This classification reveals a remarkable class of affine symmetric spaces with a non-Abelian solvable transvection group. The underlying manifold M of each element (M, ) belonging to this class is diffeomorphic to Rⁿwith the property that every tensor field on M invariant by the transvection group is constant; in particular, is not a metric connection. This classification also provides examples of nonflat affine symmetric connections on Rⁿwhich are invariant under the translations. By considering quotient spaces, one finds examples of locally affine symmetric tori which are not globally symmetric.     

Conjugate connections and Radon's theorem in affine differential geometry

For a given nondegenerate hypersurfaceM ⁿ in affine space ? ⁿ⁺¹ there exist an affine connection ?, called the induced connection, and a nondegenerate metrich, called the affine metric, which are uniquely determined. The cubic formC=?h is totally symmetric and satisfies the so-called apolarity condition relative toh. A natural question is, conversely, given an affine connection ? and a nondegenerate metrich on a differentiable manifoldM ⁿ such that ?h is totally symmetric and satisfies the apolarity condition relative toh, canM ⁿ be locally immersed in ? ⁿ⁺¹ in such a way that (?,h) is realized as the induced structure? In 1918J. Radon gave a necessary and sufficient condition (somewhat complicated) for the problem in the casen=2. The purpose of the present paper is to give a necessary and sufficient condition for the problem in casesn=2 andn≥3 in terms of the curvature tensorR of the connection ?. We also provide another formulation valid for all dimensionsn: A necessary and sufficient condition for the realizability of (?,h) is that the conjugate connection of ? relative toh is projectively flat.     

有限维非退化可解李代数的顶点算子代数

构造相应于非退化可解李代数g的顶点算子代数分两步进行,首先构造顶点代数.本文是在已经得到的相应于非退化可解李代数g的顶点代数(Vg(l,0),Y(V,1)上构造顶点算子代数.定义了非退化可解李代数g的Casimir算子Ω,给出了在伴随表示下Ω作用在g上是0及相关性质,并应用Ω定义出Vg(l,0)中元素ω,证明了Vg(l,0)关于ω的顶点算子YV(ω,x)的系数构成一个Virasoro代数-模,还证明了ω满足顶点算子代数定义中Virasoro-向量的所有公理.从而证得(Vg(l,0),Yv,1,ω)是一个顶点算子代数.     

Classification of locally rotationally symmetric Bianchi-I space–times using conformal Ricci collineations

We present a complete classification of locally rotationally symmetric (LRS) Bianchi-I space–times in accordance with their conformal Ricci collineations (CRCs). In the case where the Ricci tensor is nondegenerate, we find a general form of the vector field generating CRCs subject to some integrability conditions. Solving the integrability conditions in different cases, we find that the LRS Bianchi-I space–times admit 7-, 10-, 11-, or 15-dimensional Lie algebras of CRCs in the case where the Ricci tensor is nondegenerate. Moreover, we find that these space–times admit an infinite number of CRCs if the Ricci tensor is degenerate. We give some examples of LRS Bianchi-I space–times that admit nontrivial CRCs and are models of a perfect fluid.