The Thought Of Intrinsic Differential Geometry Created By Gauss And Its Influence

Intrinsic differential geometry plays a pivotal role in the history of the developmen-t of differential geometry. Its establishment extended the Euclidean geometry to the "benting" geometry on the surface, which became a prelude to Riemann geometry. The Intrinsic differential geometry only concerned with the nature of the objects depend on the surface itself, rather than relying on the periphery of the Euclidean space. Intrinsic differential geometry was initiated by the German mathematician Gauss. Starting in 1816, based on the geodetic work, he issued the "Disquisitiones generales circa super-ficies curvas" in 1827. In this thesis, he brilliantly described a series of new concepts and theorems of the differential geometry, put forward a great assertion that "the sur-face itself is a space", so the intrinsic differential geometry was established in one fell swoop, which developed a new field for the development of the differential geometry to higher dimensional space. Therefore, the study of the thought of Gaussian intrinsic dif-ferential geometry, is not only beneficial to dig deeper into his innovation, but also can find the source of modern differential geometry, provide historical basis for its subsequent development.Based on checking?reading and analysing of Gaussian 1827 original paper, with related to the history of the research literature. This dissertation explores the process of creation of Gaussian intrinsic differential geometry, and analyzes his wonderful intrinsic thought systematically. Thus for intrinsic differential geometry and the development of the whole differential geometry, it will provide a comprehensive understanding. The main results are as follows:1. The dissertation summarizes comprehensively the creation of intrinsic differential geometry before Gauss, in the work of three-dimensional space done by Euler?Monge and other mathematicians. Their work laid the foundation for the Gaussian intrinsic differential geometry.2. The dissertation explores systematically the process of the intrinsic differential geometry by Gauss. Based on the geodetic work, in the General investigations of curved surfaces of 1827, Gauss created the intrinsic thought, in which the theorem egregium and the Gauss-Bonnet theorem are the central ideas. What an important role these theorems play in the development of differential geometry is explained.3. The dissertation in-depth analysis of the influence of Gaussian intrinsic differential geometry thought. According to Gauss's "theorema egregium", in the middle of nine-teenth Century, the geometry studied the central problem of differential geometry—the problem of applicability, and achieved some results. Under the influence of Gauss, Rie-mann created the Riemann geometry. So the dissertation makes the important position of Gaussian intrinsic thought more clearly.