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Recently, we defined the uniruledness of symplectic manifolds and proved that symplectic uniruledness is invariant under symplectic birational

最近，我们给出了Uniruled辛流形的定义并证明了其在辛双有理变换下的不变性，为今后进一步从事辛双有理几何的研究打下了基础。

From Poisson matrix, we can observe volume-preserving vector fields. By using Geiges-Gonzalo's existence of contact circle Theorem [3] , we show the existence of volume-preserving circle on3-manifolds which says that every3-manifold admits two linear independent vector fields preserving a same volume form.

通过Poisson矩阵可以观察出保积向量场，再利用Geiges-Gonzalo的接触圆存在性定理[3]，我们证明了保积圆的存在性：每个闭的3-流形上都存在两个处处线性无关的保积向量场并且它们的常系数线性组合仍是保积向量场。

In 1943 - 1945, he worked in the Institute for Advanced Study at Princeton. Having arrived in the United States for two months, he finished his famous paper entitled "A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds", which inspired other differential geometers.

抵美两个月后，即完成其著名的论文──《闭曲面流形高斯--博内公式(Gauss-Bonnet Formula)的一个简单的内蕴证明》，对於后来微分几何的发展和微分几何学者的研究影响深远。

In these two years, cooperate with proffessor Duan Haibao, we get a height filtration on the two dimensional cohomology group of a flag manifolds. This filtration can be effectively computed.

在项目进行的两年中，我们和中科院数学所的段海豹教授合作，给出了旗流形二维上同调的Height Filtration，并给出了明确的计算。

We've found that on symplectic manifolds there exists a series of Euler-Lagrange cohomology groups different from both de Rham and harmonic cohomology in general and proposed a set of general volume-preserving equations that contain the Hamiltonian equations as a special case.

中文摘要：我们发现在作为经典力学中的相空间的辛流形上有可能存在不同于de Rham上同调和调和形式的上同调，称之为Euler-Lagrange上同调；提出相空间上保体积的一般方程，通常的正则方程是其特殊形式；进而推广了经典力学中著名的刘维定理；同时把Noether定理与上同调联系起来。

The contents include the computing of dimensions of cohomology of line bundles on non-primary Hopf manifolds and their applications to the existence problem of holomorphic structure or filtrable holomorphic structure on a continuous vector bundles; the structure and the computing of dimensions of cohomology of holomorphic vector bundles with trivial pull-back.

在这篇博士学位论文中，我们研究了非主Hopf流形上全纯向量丛，主要包括：非主Hopf流形上全纯线丛的上同调维数的计算，以及它们对连续向量丛上全纯。。。

Furthermore, under the assumption that the covariant derivative of the sectionssatisfies the Lipschitz condition with L-average along the geodesic, the estimates of the radiiof convergence balls of Newton\'s method and uniqueness balls of singular points around thesingular points of sections on Riemannian manifolds are given.

当截面的协变导数满足沿着测地线的关于取正值非减可积函数L-平均的Lipschitz条件条件时，本文给出关于截面的Newton法的收敛球半径的估计，及在奇异点附近的关于截面的奇异点的唯一性球的半径的估计。

In Riemannian manifolds, one studies Riemannian metric, covariant derivative, Riemannian connection, basic properties of the Riemann curvature tensor, curvature forms etc.

黎曼流形部分主要涉及黎曼度量，黎曼流形的定义，切向量场的协变微分，黎曼联络，黎曼几何的基本定理，曲率张量，曲率形式等概念和理论。

A pinching theorem of compact pseudoumbilical submanifolds with parallel mean curvature vector in a locally symmetric conformally flat Riemannian manifolds

研究了复空间形式中具有平行法平均曲率向量的紧致全实伪脐子流形。

The relations among three parallel sections,such as parallel mean curvature vector fields, parallel isoperimetric section and parallel umblic section,on submanifolds in the constantly curved Riemannian manifolds R ~ are studied.

讨论常曲率空间Rn+p中子流形上三类平行截面：平行平均曲率向量场、平行等参截面、平行脐截面三者之间的相互关系。

On closer examination, though, this is not a vote for multilateralism but just the opposite.

仔细审视后我们发现，这并非是对多边主义投出的赞成票，而是恰好相反。

Uncovering their weak spots, so I can defeat them.

揭露出他们的弱点，这样我就可以打败他们了。