【摘要】：微分幾何是應(yīng)用分析理論研究空間幾何性質(zhì)的一門數(shù)學(xué)學(xué)科,它與多個(gè)數(shù)學(xué)分支有密切關(guān)系,和這些學(xué)利之間相互滲透成為推動這些數(shù)學(xué)分支發(fā)展的一項(xiàng)重要工具。因此,對微分幾何的歷史發(fā)展和思想變遷進(jìn)行全面考察是十分必要的。本文以“為什么數(shù)學(xué)”為目標(biāo),對數(shù)學(xué)家成功建立數(shù)學(xué)概念、探索數(shù)學(xué)發(fā)現(xiàn)、獲取數(shù)學(xué)成果的原因進(jìn)行分析,研究歐拉的微分幾何思想、高斯的內(nèi)蘊(yùn)幾何思想根源,以及他們對后來的數(shù)學(xué)發(fā)展的深遠(yuǎn)影響。這一研究會成為微分幾何史的組成部分,也可以使我們更好地理解大地測量學(xué)作為曲面論的重要來源之一在古典微分幾何的孕育、建立和發(fā)展過程中所起的作用。取得的主要成就有：1.梳理了微分幾何的早期歷史發(fā)展,從曲線論、變分法、曲面論幾個(gè)方面闡述了歐拉對微分幾何的貢獻(xiàn)、思想根源及影響。歐拉的研究充滿了創(chuàng)新性,他引入的弧長參數(shù)、曲紋坐標(biāo)、球面映射、線元素都是內(nèi)蘊(yùn)幾何的重要元素,是適用于彎曲空間的方法和技巧。這些成果和思想豐富了微分幾何理論,為后來的微分幾何發(fā)展提供了重要的方法和思想源泉。2.剖析了高斯大地測量學(xué)中的思想和方法。高斯在漢諾威地圖繪制中使用的方法具有重大實(shí)用價(jià)值和理論價(jià)值,蘊(yùn)含著曲面理論的基本思想和方法,主要體現(xiàn)在：曲面的參數(shù)表示、弧長元素的使用、測地線的研究、局部坐標(biāo)系的建立。這些方法解釋了大地測量實(shí)踐促成高斯創(chuàng)建曲面理論的原因。3.詳細(xì)闡述了高斯1822年保角映射的論文和1827年一般曲面論的論文。這兩篇論文是內(nèi)蘊(yùn)幾何學(xué)創(chuàng)立的重要文獻(xiàn),內(nèi)容包括保角映射的一般理論、高斯絕妙定理、測地三角形內(nèi)角和定理、角度比較定理和面積比較定理等,使用的內(nèi)蘊(yùn)幾何方法有曲紋坐標(biāo)、球面映射、測地坐標(biāo)系等。高斯認(rèn)識到曲面上的幾何是局部幾何,所利用的數(shù)學(xué)工具必須有利于局部性質(zhì)的挖掘。他也注意到幾何學(xué)的中心問題是不變量的研究,在這一觀念的指引下建立了以高斯曲率為中心的內(nèi)蘊(yùn)幾何學(xué)。4.討論了高斯之后一般曲面論的補(bǔ)充和完善。通過對明金、伏雷內(nèi)等數(shù)學(xué)家著作的分析,介紹了曲面理論在19世紀(jì)的繼續(xù)發(fā)展,內(nèi)容有伏雷內(nèi)-塞克雷公式、測地曲率、曲面論基本方程、曲面的存在性定理、曲面的可貼合性等。曲紋坐標(biāo)、第一基本形式、標(biāo)架等內(nèi)蘊(yùn)幾何工具得到了普遍使用,內(nèi)蘊(yùn)幾何思想得到了廣泛傳播和深刻領(lǐng)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.
【學(xué)位授予單位】：西北大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2016
【分類號】：O186.1

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