Some Research Of Th E History Of Manifold And Related Theories

Manifold concept originated in a presentation on geometry basis lectured by German mathematician Riemann in 1854, in which he divided manifold theory into two parts: geometry and topology. Subsequently mathematicians began to study manifold along these two directions respectively and obtained a lot of results. However, the strict de?nition of manifold had not been got yet, which hindered the further development of this discipline. In 1913, Weyl’s publication Die Idee der Riemannschen Fl?che ?rst gave two-dimensional manifold an axiomatic de?nition, then manifold theory entered a new period.By the mid-20 th century, manifold became the foundation of differential geometry,differential topology, global analysis, differential dynamic system and foliation, etc. These subjects belong to structural mathematics category and are in the mainstream of the development of modern mathematics. It can be said that manifold is a representative concept and theorey in the 20 th century mathematics, manifold has become one of the most important ideas of modern mathematics, and plays an increasingly important role in mathematics as well as theoretical physics.With original literature in hand and related research literature supplemented, according to the chronology and the works of important mathematicians, this dissertation sorts and summarizes the historical origins and theoretical framework of manifold; explores early mathematicians’ views on manifold, such as Riemann, Klein and Poincaré,investigates the contribution of later mathematiciansto manifold, such as Weyl, Veblen and Whitney. The main contents are as follows:1. Sort and summarize the overall framework of the development of manifold from the 1850 s to the 1930 s.2. Explore the origins of manifold concept by the appearance or loom of mianfoldfrom geometry, analytics and physics aspects. Analyse Riemann n-extended manifold, point out local Euclidean and differentiable are two features of which, discuss the concept of curvature of n-extended manifold and the in?uence of Riemann’s presentation.3. Explore Klein’s academic background for the ?rst time, and investigate Clifford andprym’sin?uenceonhisunderstandingmanifold.focusonklein’sviewonmani-fold,introducethemaincontentsoferlangenprogrammeandüberriemannstheoriederalgebraischenfunktionenundihrerintegrale.askleinhaddifferentobjectives,hisunderstandingandtreatmentofmanifoldweredifferentfromriemann’s.4.examinetheconceptofmanifoldinanalysisstiusandits?vesupplementsofpoincaré,introducepoincaré’stwomethodsofde?ningandrepresentingmanifold,analysetheessenceandrelationsofthetwode?nitions,interpretgeometricrepresentationanddiscontinuousgrouprepresentationofmanifold.discussthelifeandworkofthedanishmathematicianp.heeggardroughly.inaddition,introducethedevelopmentoftopologyandtopologistsafterpoincaréinacertaindegree.5.basingontheoriginalliterature,introducethemaincontents,featuresandin-?uencesofdieideederriemannschenfl?che.analyseweyl’spurpose,motivationandmeansofintroducingtheconceptofmanifold,sumuptheclueofmanifoldintroduction,explorethein?uenceofklein,hilbertandothersonweyl.analyseweyl’scontributiontotheconceptsofmanifoldandriemannsurfacein1913historically,anddiscusshismathematicalphilosophybrie?y.6.dodetailedbiographyresearchonamericanmathematiciansveblenandwhitney,interpretveblen’smanifoldde?nitionandwhitney’sembeddingtheorem,analysetheircontributiontomanifoldhistoricallyfromaxiomaticmanifoldperspective.7.studythechinesetranslationofmanifold,highlyevaluatejiangzehan’svalida-tionworkontopologicalterms.