![]() It is not too far-fetched to argue that differential geometry should be in every mathematician's arsenal. The field has even found applications to group theory as in Gromov's work and to probability theory as in Diaconis's work. Differential geometry is also useful in topology, several complex variables, algebraic geometry, complex manifolds, and dynamical systems, among other fields. Over the past one hundred years, differential geometry has proven indispensable to an understanding of the physical world, in Einstein's general theory of relativity, in the theory of gravitation, in gauge theory, and now in string theory. It dates back to Newton and Leibniz in the seventeenth century, but it was not until the nineteenth century, with the work of Gauss on surfaces and Riemann on the curvature tensor, that differential geometry flourished and its modern foundation was laid. How can we use discrete differential geometry to solve problems in computer graphics, geometry processing, and beyond In this paper, Keenan Crane, a leading researcher and professor at CMU, introduces the main concepts and applications of this exciting field, with a focus on discrete exterior calculus and discrete differential forms. Additionally, in an attempt to make the exposition more self-contained, sections on algebraic constructions such as the tensor product and the exterior power are included.ĭifferential geometry, as its name implies, is the study of geometry using differential calculus. ![]() By studying the properties of the curvature of curves on a sur face, we will be led to the rst and second fundamental forms of a surface. For the benefit of the reader and to establish common notations, Appendix A recalls the basics of manifold theory. differential geometry and about manifolds are refereed to doCarmo12,Berger andGostiaux4,Lafontaine29,andGray23.Amorecompletelistofreferences can be found in Section 20.11. Prerequisite material is contained in author's text An Introduction to Manifolds, and can be learned in one semester. The DifferentialGeometry Software Project uses the Maple Mathematics engine to symbolically perform fundamental operations of calculus on manifolds, differential geometry, tensor calculus, Lie algebras, Lie groups, transformation groups, jet spaces, and variational calculus. A knowledge of de Rham cohomology is required for the last third of the text. Maple Francis Wright Linear Algebra: A geometric approach E. After the first chapter, it becomes necessary to understand and manipulate differential forms. ![]() Initially, the prerequisites for the reader include a passing familiarity with manifolds. ![]() Exercises throughout the book test the reader’s understanding of the material and sometimes illustrate extensions of the theory. The exposition follows the historical development of the concepts of connection and curvature with the goal of explaining the Chern–Weil theory of characteristic classes on a principal bundle. Along the way we encounter some of the high points in the history of differential geometry, for example, Gauss' Theorema Egregium and the Gauss–Bonnet theorem. This text presents a graduate-level introduction to differential geometry for mathematics and physics students. ![]()
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