Differential Geometry

Abstract

A branch of geometry dealing with geometrical forms, mainly with curves and surfaces, by methods of mathematical analysis. In di erential geometry the properties of curves and surfaces are usually studied on a small scale, i.e. the study concerns properties of su ciently small pieces of them. Properties of families of curves and surfaces are also studied. Di erential geometry arose and developed in close connection with mathematical analysis, the latter having grown, to a considerable extent, out of problems in geometry. Many geometrical concepts were de ned prior to their analogues in analysis. For instance, the concept of a tangent is older than that of a derivative, and the concepts of area and volume are older than that of the integral. Di erential geometry rst appeared in the 18th century and is linked with the names of L. Euler and G. Monge. The rst synoptic treatise on the theory of surfaces was written by Monge (Une application d'analyse a la g eom etrie, 1795). In 1827 a study under the (English) title A general study on curved surfaces was published by C.F. Gauss; this study laid the foundations of the theory of surfaces in its modern form. From that time onwards di erential geometry ceased to be a mere application of analysis, and has become an independent branch of mathematics. The discovery of non-Euclidean geometry by N.I. Lobachevskii in 1826 played a major role in the development of geometry as a whole, including di erential geometry. Lobachevskii rejected in fact the a priori concept of space, which was predominating in mathematics and in philosophy. He found that spaces di erent from Euclidean spaces exist. This idea of Lobachevskii was re ected in numerous mathematical studies. Thus, in 1854 B. Riemann published his course ber die Hypothesen, welche der Geometrie zuGrunde liegen and thus laid the foundations of Riemannian geometry, the application of which to higher-dimensional manifolds is related to the geometry of n-dimensional space similarly as the relation between the interior geometry of a surface and Euclidean geometry on a plane. Historically, di erential geometry arose and developed as a result of and in