The Development Of Intrinsic Differential Geometry From Gauss To Riemann

The development of intrinsic differential geometry from Gauss to Riemann represents the transition of differential geometry from classical to modern,which is a crucial change in the history of differential geometry.Based on original literature and related research literature,guided by “why mathematics” ideology and “reception history” methodology,this dissertation studies the history of instrinsic differential geometry from Gauss to Riemann.Through a different viewing angle from previous studies,this dissertation studies not only Gauss and Riemann’s intrinsic differential geometrical work,but also the reception history of Gauss’ s intrinsic differential geometry before Riemann.In addition,the origin of some core concept in differential geometry is investigated,including the first and second fundamental form,total curvature and geodesic curvature.The main research results obtained are as follows:1.The history of differential geometry before Gauss is studied by this dissertation.This dissertation investigated the problem of the origin of differential geometry and earliest differential geometrical work,and the history of curve theory and surface theory before Gauss.The differential geometrical work of Euler,Monge,Meusnier and Lagrange on curvature of surface,developable surface and minimal surface are elaborated.The instrument of curvilinear coordinate and line element used by Euler in the problems of developable surface later become fundamental tools of Gauss in his intrinsic differential geometry.2.This dissertation investigated the process from origin to maturity of Gauss’ s intrinsic differential geometrical thought.Through researching on relevant articles and manuscripts in Gauss’ s collected works,this dissertation tries to recover the establishment process of Theorema Egregium and construct the logical line of this process,which represents the formation of intrinsic differential thought.By systematically interpretation of the manuscript of 1825 and the article of 1828,Gauss’ s original points in his work is summarized and the deepening process of his intrinsic differential geometrical thought is revealed.3.This dissertation studied the reception and early development of Gauss’ s intrinsic differential geometry before Riemann.Through thorough interpretation of Minding’s original work on the problems of development of surfaces and geodesic curvature,Minding’s contribution to instrinsic differential geometry is summarized,and several concret propositions by Minding which hasn’t been mentioned by previous studies are discovered.This dissertation also systematically investigated the reception process of intrinsic differential geometry in France,and the contribution of intrinsic differential geometry of Liouville and Bonnet.The history of fundamental equations and fundamental theorem of surface theory and Peterson’s contribution to them is studied.4.Riemann’s probationary lecture on the foundation of geometry is thoroughly interpretated.From three aspects of space philosophy,non-Euclidean geometry and intrinsic differential geometry,the background of the problem of the foundation of geometry is investigated,the lecture is interpreted and its contribution in history is analyzed.This dissertation also summarized the relationship and the distinction between Riemann and Gauss’ s differential geometry.