【作者】 张楠；

【导师】 刘会立；

【作者基本信息】 东北大学 ， 基础数学， 2008， 硕士

【摘要】 微分几何是一门历史悠久的学科.甚至可以这样说,在微积分诞生的同时就诞生了微分几何,不过这门学科的生命力至今仍很旺盛.近年来它对其他分支的影响也越来越深刻,对于自然学科中其他学科的影响范围也越来越广.与此同时这门学科本身从内容到方法也在不断更新.曲线论与曲面论是微分几何中两大重要内容,其中直纹面由于具有很好的性质在曲面论中占据十分重要的地位,受到曲线论中伴随曲线对(如Bertrand曲线对,Mannheim曲线对)思想的启发,联想到研究曲面论中直纹面及其伴随曲面的对应关系问题.三维欧氏空间中的直纹面可以是可展曲面,也可以是极小曲面,单叶双曲面和双曲抛物面.本文用微分几何的方法,从三维欧氏空间中的直纹面出发,分别讨论以上四类直纹面的伴随曲面的性质.第一章介绍本文的研究背景及相关的基础知识,如欧氏空间、直纹面、可展曲面、极小曲面、单叶双曲面、双曲抛物面的定义和性质等.第二章对一般形式的直纹面及其伴随曲面的各个基本量,高斯曲率及平均曲率进行对比和分析.第三章研究三类可展曲面的伴随曲面.第四章讨论两类极小直纹面的伴随曲面.第五章分别讨论单叶双曲面和双曲抛物面的伴随曲面.最后一章对本文工作进行总结与展望.更多还原

【Abstract】 Differential geometry has a long history. Calculus and differential geometry were born almost at the same time. Recent years, the effect of differential geometry on other branches is more and more profound. The scope it has influenced on other subjects in natural subject is also increasingly expanding. At the same time this subject itself is updated constantly. Curve theory and surface theory are two important elements in differential geometry. Ruled surface which has good characters occupies very important position in surface theory. Associating with the adjoint pair of curves, such as Bertrand pair of curves and Mannheim pair of curves, it is natural to study the corresponding relation between ruled surface and its adjoint surface in surface theory.Ruled surface in 3-dimensional Euclidean space can be developable surface, minimum surface, unparted hyperboloid and hyperbolic paraboloid. Using the method of differential geometry, the adjoint surfaces of four kinds of ruled surfaces that have been mentioned above are discussed in this paper. In chapter one, the background and the basic knowledge related are introduced, such as the definition and character of Euclidean space, ruled surface, developable surface and minimum surface. In chapter two, the general form of ruled surface and its adjoint surface are compared with the various fundamental quantity, Gaussian curvature and mean curvature. In chapter three, three types of adjoint surface of developable surface are studied. In chapter four, two types of adjoint surface of minimum ruled surface are discussed. In chapter five, the adjoint surface of unparted hyperboloid and hyperbolic paraboloid are also discussed respectively. The last chapter is the summary and expect of the paper.更多还原