differential geometry curves and surfaces

The essential mathematical object is that of a regular surface. The study of this field, which was initiated in its modern form in the 1700s, has led to the development of higher-dimensional and abstract geometry, such as Riemannian geometry and general relativity. The differential geometry of surfaces is concerned with a mathematical understanding of such phenomena. It is intuitively quite familiar to say that the leaf of a plant, the surface of a glass, or the shape of a face, are curved in certain ways, and that all of these shapes, even after ignoring any distinguishing markings, have certain geometric features which distinguish one from another. Darboux collected many results in his four-volume treatise Théorie des surfaces (1887–1896).įor broader coverage of this topic, see Surface (mathematics). The nineteenth century was the golden age for the theory of surfaces, from both the topological and the differential-geometric point of view, with most leading geometers devoting themselves to their study. This point of view was extended to higher-dimensional spaces by Riemann and led to what is known today as Riemannian geometry. The crowning result, the Theorema Egregium of Gauss, established that the Gaussian curvature is an intrinsic invariant, i.e. This marked a new departure from tradition because for the first time Gauss considered the intrinsic geometry of a surface, the properties which are determined only by the geodesic distances between points on the surface independently of the particular way in which the surface is located in the ambient Euclidean space.

The defining contribution to the theory of surfaces was made by Gauss in two remarkable papers written in 18.

Monge laid down the foundations of their theory in his classical memoir L'application de l'analyse à la géometrie which appeared in 1795. In 1760 he proved a formula for the curvature of a plane section of a surface and in 1771 he considered surfaces represented in a parametric form. Curvature of general surfaces was first studied by Euler. The development of calculus in the seventeenth century provided a more systematic way of computing them. The volumes of certain quadric surfaces of revolution were calculated by Archimedes. This is well illustrated by the non-linear Euler–Lagrange equations in the calculus of variations: although Euler developed the one variable equations to understand geodesics, defined independently of an embedding, one of Lagrange's main applications of the two variable equations was to minimal surfaces, a concept that can only be defined in terms of an embedding.

On the other hand, extrinsic properties relying on an embedding of a surface in Euclidean space have also been extensively studied. These Lie groups can be used to describe surfaces of constant Gaussian curvature they also provide an essential ingredient in the modern approach to intrinsic differential geometry through connections. An important role in their study has been played by Lie groups (in the spirit of the Erlangen program), namely the symmetry groups of the Euclidean plane, the sphere and the hyperbolic plane. Surfaces naturally arise as graphs of functions of a pair of variables, and sometimes appear in parametric form or as loci associated to space curves. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space. Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.

Differential geometry curves and surfaces