For a point on a curve defined by the general equation (1) to be regular, it is necessary and sufficient that the inequality In differential geometry the study of a curve mainly concerns a neighbourhood of a regular point. The curve can be defined, in a neighbourhood of that point, by equations of the formĪre differentiable functions. A point on a curve is said to be regular if, by a suitable choice of a Cartesian coordinate system $ x, y, z $, Of these, the so-called natural parametrization, in which the length of an arc of the curve, counted from some given point, serves as the parameter, is especially important. The degree of differentiability of the curve is given by the degree of differentiability of $ x ( t), y ( t) $įor one and the same curve there are uncountably many ways of parametric description of the type (1). These are curves which can locally be specified by equations of the typeĪre sufficiently regular functions of a parameter $ t $. The major subject of the theory of curves are the so-called differentiable curves. 2 The theory of surfaces and its generalizations.These studies were continued by many Russian and Soviet geometers. such continuous deformations of surfaces during which the interior geometry remains invariant. Deformation, isometric) of surfaces, i.e. Their studies largely concern the problems of isometric deformation (cf. In Russia a school of differential geometry was established by F. Affine connection Projective connection). Cartan established the theory of spaces with projective connections and affine connections (cf. Cartan in relation to differential geometry. that geometry is the study of invariants under groups of transformations - was developed by E. Klein in 1872 in his Erlangen program - viz. Riemann published his course Über 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 $-ĭimensional space similarly as the relation between the interior geometry of a surface and Euclidean geometry on a plane. This idea of Lobachevskii was reflected in numerous mathematical studies. He found that spaces different from Euclidean spaces exist. Lobachevskii rejected in fact the a priori concept of space, which was predominating in mathematics and in philosophy. Lobachevskii in 1826 played a major role in the development of geometry as a whole, including differential geometry. The discovery of non-Euclidean geometry by N.I.
From that time onwards differential geometry ceased to be a mere application of analysis, and has become an independent branch of mathematics.
Gauss this study laid the foundations of the theory of surfaces in its modern form. In 1827 a study under the (English) title A general study on curved surfaces was published by C.F. The first synoptic treatise on the theory of surfaces was written by Monge (Une application d'analyse à la géométrie, 1795). 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.ĭifferential geometry first appeared in the 18th century and is linked with the names of L. Many geometrical concepts were defined prior to their analogues in analysis. Properties of families of curves and surfaces are also studied (see, for example, Congruence Web).ĭifferential geometry arose and developed in close connection with mathematical analysis, the latter having grown, to a considerable extent, out of problems in geometry.
the study concerns properties of sufficiently small pieces of them. In differential geometry the properties of curves and surfaces are usually studied on a small scale, i.e. A branch of geometry dealing with geometrical forms, mainly with curves and surfaces, by methods of mathematical analysis.
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