Research On The Theory Of Gaussian Intrinsic Differential Geometry

Differential Geometry is a mathematical discipline which uses techniques of mathematical analysis to study problems in geometry. It has a close relation with many mathematical branches and all of these branches interpenetrate and interact. Therefore, it is necessary to investigate the history and thought of differential geometry comprehensively. Taking "why mathematics is done" as the starting point and main purpose and analyzing the origin of mathematical concepts, theorems and results, the dissertation studies the theory of intrinsic geometry of Euler and Gauss, as well as its profound influence on geometric development. The study is a component part of the history of differential geometry and illustrates the role of geodesy in the establishment and advancement of classical differential geometry. The main contributions of this study are as follows:1. The early history of differential geometry, especially Euler's contribution is expounded. There are three aspects in Euler's research:space curves, calculus of variations, curved surfaces. His differential geometry included many creative techniques:arc length, curvilinear coordinate, spherical representation, line element, which are appropriate tools for curved spaces and belong to intrinsic geometry. Euler's research not only enriched the theory of differential geometry, but also provided the sources of thought and methods.2. Gauss's geometric thought in geodesy is elaborated. There are important theoretical value and practical value in producing a geodesic map of Duchy of Hannover. The methods in the mapping of Hannover, such as parametric equation of a surface, line element, geodesic line, local coordinates, contain basic thought and technique of differential geometry, which explains the reason why Gauss's differential geometry originates from his geodesic practice.3. Gauss's two famous papers in 1822 and in 1827 are explained. Both of the papers, which are important to the establishment of intrinsic differential geometry, contain conforming maps, Gauss great theorem, the theorem of the sum of the angles of a geodesic triangle, theorem of comparison of angles, theorem of comparison of areas, as well as representative methods in intrinsic geometry. Gauss knew geometry on curved surfaces was local geometry and mathematical tools should fit for the exploration of local characters. He was also aware that the central problem in geometry was invariants which lead to intrinsic geometry centering around Gaussian curvature.4. Complement and perfection of theory of surface after Gauss is discussed. The development of intrinsic differential geometry in 19 century is introduced through works of Minding, Frenet and so on. The content includes Frenet-Serret formulas, geodesic curvature, the fundamental equations of surfaces, theorem of existence of surfaces, applicability of surfaces and so on. Curvilinear coordinate, the first fundamental form, the moving trihedral were popularized in intrinsic geometry and the thought of intrinsic differential geometry was spread widely and understood deeply.