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And then, in this part, following the idea of Topping [51, 52], we will show an isoperimetric inequality (6.3.1) on surfaces of non-positive Gaussian curvature by means of the curve shortening flow on surfaces, this inequality can be considered as a generalization of the Banchoff-Pohl inequality in the Euclidean plane.

然后，仿照Topping[51]和[52]利用平均曲率流证明等周不等式的想法，我们利用曲面上的曲线缩短流在Gauss曲率非正的曲面上证明了一个等周不等式(6.3.1)，它可以看成欧氏平面上Banchoff-Pohl [8]不等式在曲面上的推广。

This paper gives the Euler characteristic of 2-dimemsional surfaces with crosscaps and surfaces with boundaries.

本文给出了带有交叉帽的二维曲面和带有边缘边的二维曲面的欧拉示性数。

Cloths are safe to use on many surfaces effectively to remove tarnish, grime, and fingerprints on silver, gold, copper, bronze, flatware, brass lamps, tile, hardwood and many other surfaces.

布是安全有效地使用在许多表面，以消除损害，污垢，指纹和银，金，铜，青铜，餐具，铜灯，瓷砖，硬木和许多其他表面。

In this paper we discuss the existence of surfaces of rotation about given Gauss curvature function in R 3 and give the position vector field of these surfaces.

讨论了R3 内给定 Gauss曲率函数的旋转曲面的存在性问题并且给出了这类曲面的位置向量场。

In this paper, we discuss the existence problem of rotation surfaces of given Gauss curvature function in H_1~3(-1) and give the position vector field of the surfaces.

中文摘要：本文讨论了三维反de sitter空间H_1~3(-1)中给定Gauss曲率函数的旋转曲面M的存在性问题并给出了这类旋转曲面的位置向量场。

In this paper, we mainly discuss a kind of surfaces in4- dimensional Pseudo-Euclidean Space and give the classification of such surfaces with constant mean curvature and Gauss curvature.

文章研究四维伪欧氏空间中的一类曲面，给出了具有常平均曲率、高斯曲率及相关平均曲率和高斯曲率的这类曲面的分类。

In section three, we mainly discuss the existence problemof rotation surfaces of given Gauss curvature function in H_1~3(-1) and give theposition vector field of the surfaces and obtain theorem one and theorem two.

第三节主要讨论了H_1~3(-1)中给定Gauss曲率函数的旋转曲面M的存在性问题并给出了这类旋转曲面的位置向量场，得到了定理1和定理2。

Can be used on any open gearing, sliding surfaces, and on specific bearing surfaces.

可用于任何打开的齿轮，滑动面，并就具体轴承的表面。

Based on the theory of gearing and differential geometry, a series of geometric relationships between conjugate tooth surfaces are analyzed, basic relationships of normal curvature as well as geodesic torsion between conjugate tooth surfaces are derived mathematically for the most general form of gearing.

基于微分几何和齿轮啮合理论，分析了共轭啮合齿面间的一系列几何关系，从而揭示了两个作空间共轭啮合运动的齿面之间的法曲率、短程挠率的内在几何联系。作为上述理论的具体应用，研究了准双曲面齿轮传动两齿面之间在给定参考点处的法曲率、短程挠率的关系

Based on the theory of gearing and differential geometry, a series of geometric relationships between conjugate tooth surfaces are analyzed, basic relationships of normal curvature as well as geodesic torsion between conjugate tooth surfaces are derived mathematically for the most general form of gearing .

作为上述理论的具体应用，研究了准双曲面齿轮传动两齿面之间在给定参考点处的法曲率、短程挠率的关系。

Neither the killing of Mr Zarqawi nor any breakthrough on the political front will stop the insurgency and the fratricidal murders in their tracks.

在对危险的南部地区访问时，他斥责什叶派民兵领导人对中央集权的挑衅行为。

In fact,I've got him on the satellite mobile right now.

实际上 我们已接通卫星可视电话了