高斯的内蕴微分几何理论研究

发布时间：2018-07-23 09:43

【摘要】：微分几何是应用分析理论研究空间几何性质的一门数学学科,它与多个数学分支有密切关系,和这些学利之间相互渗透成为推动这些数学分支发展的一项重要工具。因此,对微分几何的历史发展和思想变迁进行全面考察是十分必要的。本文以“为什么数学”为目标,对数学家成功建立数学概念、探索数学发现、获取数学成果的原因进行分析,研究欧拉的微分几何思想、高斯的内蕴几何思想根源,以及他们对后来的数学发展的深远影响。这一研究会成为微分几何史的组成部分,也可以使我们更好地理解大地测量学作为曲面论的重要来源之一在古典微分几何的孕育、建立和发展过程中所起的作用。取得的主要成就有：1.梳理了微分几何的早期历史发展,从曲线论、变分法、曲面论几个方面阐述了欧拉对微分几何的贡献、思想根源及影响。欧拉的研究充满了创新性,他引入的弧长参数、曲纹坐标、球面映射、线元素都是内蕴几何的重要元素,是适用于弯曲空间的方法和技巧。这些成果和思想丰富了微分几何理论,为后来的微分几何发展提供了重要的方法和思想源泉。2.剖析了高斯大地测量学中的思想和方法。高斯在汉诺威地图绘制中使用的方法具有重大实用价值和理论价值,蕴含着曲面理论的基本思想和方法,主要体现在：曲面的参数表示、弧长元素的使用、测地线的研究、局部坐标系的建立。这些方法解释了大地测量实践促成高斯创建曲面理论的原因。3.详细阐述了高斯1822年保角映射的论文和1827年一般曲面论的论文。这两篇论文是内蕴几何学创立的重要文献,内容包括保角映射的一般理论、高斯绝妙定理、测地三角形内角和定理、角度比较定理和面积比较定理等,使用的内蕴几何方法有曲纹坐标、球面映射、测地坐标系等。高斯认识到曲面上的几何是局部几何,所利用的数学工具必须有利于局部性质的挖掘。他也注意到几何学的中心问题是不变量的研究,在这一观念的指引下建立了以高斯曲率为中心的内蕴几何学。4.讨论了高斯之后一般曲面论的补充和完善。通过对明金、伏雷内等数学家著作的分析,介绍了曲面理论在19世纪的继续发展,内容有伏雷内-塞克雷公式、测地曲率、曲面论基本方程、曲面的存在性定理、曲面的可贴合性等。曲纹坐标、第一基本形式、标架等内蕴几何工具得到了普遍使用,内蕴几何思想得到了广泛传播和深刻领悟。
[Abstract]:Differential geometry is a mathematical subject which studies the properties of spatial geometry by applying analytical theory. It is closely related to many branches of mathematics, and the infiltration between these branches of learning and profit has become an important tool to promote the development of these branches of mathematics. Therefore, it is necessary to investigate the historical development and ideological changes of differential geometry. This paper aims at "Why Mathematics", analyses the reasons why mathematicians have successfully established mathematical concepts, explored mathematical discoveries, obtained mathematical achievements, and studied Euler's idea of differential geometry and the origin of Gao Si's thought of intrinsic geometry. And their profound influence on later mathematical developments. This study will become an integral part of the history of differential geometry and will help us better understand the role of geodesy as one of the important sources of surface theory in the gestation, establishment and development of classical differential geometry. The main achievements are: 1: 1. This paper reviews the early historical development of differential geometry, and expounds Euler's contribution to differential geometry, ideological origin and influence from the following aspects: curve theory, variational method and surface theory. Euler's research is full of innovation. The arc length parameters, curved coordinates, spherical mapping and line elements are all important elements of intrinsic geometry, which are the methods and techniques suitable for bending space. These results and ideas enrich the theory of differential geometry and provide an important method and source of thought for the later development of differential geometry. The ideas and methods of Gao Si geodesy are analyzed. The method used by Gao Si in Hannover map drawing has great practical and theoretical value, and contains the basic ideas and methods of surface theory, which are mainly embodied in: surface parameter representation, the use of arc length elements, the study of geodesic. The establishment of local coordinate system. These methods explain why geodetic practice contributed to the creation of surface theory by Gao Si. The paper of conformal mapping of Gao Si 1822 and general surface theory of 1827 are expounded in detail. These two papers are important documents of intrinsic geometry, including the general theory of conformal mapping, Gao Si's excellent theorem, geodesic triangle interior angle sum theorem, angle comparison theorem and area comparison theorem, etc. The methods used include curved coordinates, spherical maps, geodesic coordinates, and so on. Gao Si recognizes that geometry on a surface is a local geometry, and the mathematical tools used must be beneficial to the mining of local properties. He also noted that the central problem of geometry is the study of invariants, and under the guidance of this concept, an intrinsic geometry with Gao Si curvature as its center was established. 4. The supplement and perfection of general surface theory after Gao Si are discussed. Based on the analysis of the works of mathematicians such as Minkin and Vorne, this paper introduces the development of surface theory in the 19th century. The contents include Fline-Seckley formula, geodesic curvature, the basic equation of surface theory, and the existence theorem of surface. The compatibility of surfaces, etc. Curved coordinates, the first basic form, frame and other intrinsic geometric tools have been widely used, the idea of intrinsic geometry has been widely spread and deeply understood.
【学位授予单位】：西北大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O186.1

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