局部共形对称黎曼流形的孤立现象

讨论了局部共形对称的封闭黎曼流形,证明了黎曼曲率张量模长的一个拼挤定理.当M是局部共形平坦流形时,得到了曲率张量模长的最佳拼挤常数,并确定了达到该值的黎曼流形.     

具有负数量曲率的紧致黎曼流形的Killing向量场

本文研究了具有负数量曲率的紧致黎曼流形上的Killing向量场.利用Bochner方法,得到在此类流形上非平凡的Killing向量场的存在的必要条件.这个结果拓广了文献[6]中的定理1.     

四元数空间形式中全复子流形的内蕴积分等式

本文给出了四元数空间形式中全复子流形的一个性质．即设Ｍ２ｎ是的全复子流形，ρ，‖Ｒｉｅｍ‖２，‖Ｒｉｃｃｉ‖２分别表示Ｍ２ｎ的纯量曲率和黎曼曲率，Ｒｉｃｃｉ曲率的模长平方，则在Ｍ上处处成立．     

径向截面曲率有下界黎曼流形的微分同胚定理

研究了径向截面曲率以一类旋转模曲面的Gauss曲率为下界的非紧完备黎曼流形的拓扑,得到了该类黎曼流形与欧氏空间微分同胚的一个合理的充分条件,推广了径向截面曲率有常数下界完备黎曼流形的微分同胚定理.     

李奇曲率平行的黎曼流形的曲率张量模长

李安民和赵国松［１］提出了下面的问题：找出李奇曲率平行的黎曼流形的曲率张量模长的最佳拼挤常数并确定达到该值的流形．本文确定了非爱因斯坦流形的最佳拼挤常数和达到该值的黎曼流形．在ｎ１２时，回答了［１］中提出的问题．     

具有两个不同Ricci主曲率的局部共形平坦Riemann流形

得到了具有常m阶Schouten曲率与两个不同Schouten主曲率(或者等价地,两个不同Ricci上曲率)的完备局部共形平坦Riemann流形的分类结果.作为应用,得到了若干Schouten张量的pinching性质.     

S~n中子流形的Moebius特性

研究Sn中不舍脐点且Moebius形式为零的子流形的Moebius特性．首先得到子流形的Moebius标准数量曲率与截面曲率的一个关系定理,然后分别利用迹为零的Blaschke张量、Moebius标准数量曲率、截面曲率所满足的某种内蕴关系刻画了驴中子流形的Moebius特性．     

微分形式的Liouville型定理

本文建立了具有有界的负截面曲率的完备单连通黎曼流形上 .其应力能量张量守恒的 L2 -形式的一个消失定理 .从而推广了忻元龙的新近结果 ,给出了 Dodziuk猜想的部分回答     

具有两个不同Ricci主曲率的局部共形平坦Riemann流形

得到了具有常m阶Schouten曲率与两个不同Schouten主曲率(或者等价地,两个不同Ricci主曲率)的完备局部共形平坦Riemann流形的分类结果.作为应用,得到了若干Schouten张量的pinching性质.     

局部共形对称黎曼流形上的保圆曲率张量长度的整体刚性定理

本文我们研究了局部共形对称闭黎曼流形, 建立了一个关于保圆曲率矢量长度的整体刚度定理.     

~2Ric=0的Riemann流形的刚性定理(英文)

本文用Ｒｉｃ表示里奇曲率张量，研究了２Ｒｉｃ＝０的黎曼流形什么时候成为爱因斯坦流形或空间形式     

▽2Ric=0的Riemann流形的刚性定理

本文用Ｒｉｃ表示里奇曲率张量，研究了▽²Ｒｉｃ＝０的黎曼流形什么时候成为爱因斯坦流形或空间形式     

具有调和曲率与有限$L^p$曲率的完备流形

Let(M~n, g)(n ≥ 3) be an n-dimensional complete Riemannian manifold with harmonic curvature and positive Yamabe constant. Denote by R and R?m the scalar curvature and the trace-free Riemannian curvature tensor of M, respectively. The main result of this paper states that R?m goes to zero uniformly at infinity if for p ≥ n, the L~p-norm of R?m is finite.As applications, we prove that(M~n, g) is compact if the L~p-norm of R?m is finite and R is positive, and(M~n, g) is scalar flat if(M~n, g) is a complete noncompact manifold with nonnegative scalar curvature and finite L~p-norm of R?m. We prove that(M~n, g) is isometric to a spherical space form if for p ≥n/2, the L~p-norm of R?m is sufficiently small and R is positive.In particular, we prove that(M~n, g) is isometric to a spherical space form if for p ≥ n, R is positive and the L~p-norm of R?m is pinched in [0, C), where C is an explicit positive constant depending only on n, p, R and the Yamabe constant.     

Expression of curvature tensors and some applications

We get an explicit expression of curvature operators in terms of at most eight terms of sectional curvatures. Some applications of this result are also given, particularly we improve a result of Chen-Tian related to the first Chern class of admissible surfaces in pinched manifolds. We also characterize in a simple way all functionsk(x, y) which can be sectional curvatures of some curvature operatorR.Supported by CNPq, Brazil and NNSFC.     

Finsler空间上的Weyl曲率

The Weyl curvature of a Finsler metric is investigated. This curvature constructed from Riemannain curvature. It is an important projective invariant of Finsler metrics. The author gives the necessary conditions on Weyl curvature for a Finsler metric to be Randers metric and presents examples of Randers metrics with non-scalar curvature. A global rigidity theorem for compact Finsler manifolds satisfying such conditions is proved. It is showed that for such a Finsler manifold,if Ricci scalar is negative,then Finsler metric is of Randers type.     

Complete noncompact manifolds with harmonic curvature

Let (M ⁿ , g) be an n-dimensional complete noncompact Riemannian manifold with harmonic curvature and positive Sobolev constant. In this paper, by employing an elliptic estimation method, we show that (M ⁿ , g) is a space form if it has sufficiently small L ^n/2-norms of trace-free curvature tensor and nonnegative scalar curvature. Moreover, we get a gap theorem for (M ⁿ , g) with positive scalar curvature.     

Stability of the p-Curvature Positivity under Surgeries and Manifolds with Positive Einstein Tensor

We establish the stability of the class of manifolds with positive p-curvature under surgeries in codimension p + 3. As a consequence of this result, we first obtain the classification of compact 2-connected manifolds of dimension 7 with positive Einstein tensor; and secondly the existence of metrics with positive Einstein tensor on any compact, simply connected, non-spin manifold of dimension 7 whose second homotopy group is isomorphic to Z₂.     

Fedosov supermanifolds

We study the properties of the symplectic curvature tensor on supermanifolds. Using the method of normal coordinates, we establish higher-order relations between affine extensions of the curvature tensor and of the symplectic structure tensor. We find a generating function for higher-order relations. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 2, pp. 202–227, November, 2006.