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. ![]() 11 Global differential geometry of surfaces.10 Riemannian connection and parallel transport. ![]() 4.5 Surfaces of constant Gaussian curvature.3.4 Christoffel symbols, Gauss–Codazzi equations, and the Theorema Egregium.3.3 First and second fundamental forms, the shape operator, and the curvature.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. Louis Kauffman, University of Illinois at ChicagoIn mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. ![]() This book provides a good, often exciting and beautiful basis from which to make explorations into this deep and fundamental mathematical subject. … There are many points of view in differential geometry and many paths to its concepts. ![]() This book on differential geometry by Kühnel is an excellent and useful introduction to the subject. In addition to a variety of improvements, the author has included solutions to many of the problems, making the book even more appropriate for use in the classroom. This new edition is an improved version of what was already an excellent and carefully written introduction to both differential geometry and Riemannian geometry. This new edition provides many advancements, including more figures and exercises, and-as a new feature-a good number of solutions to selected exercises. The prerequisites are undergraduate analysis and linear algebra. The text is illustrated with many figures and examples. The second part studies the geometry of general manifolds, with particular emphasis on connections and curvature. The first half covers the geometry of curves and surfaces, which provide much of the motivation and intuition for the general theory. This carefully written book is an introduction to the beautiful ideas and results of differential geometry.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |