Differential Geometry: Manifolds, Curves, and Surfaces [electronic resource]

Differential Geometry: Manifolds, Curves, and Surfaces
Author: Marcel Berger, Bernard Gostiaux
Published by Springer New York
ISBN: 978-1-4612-6992-2
DOI: 10.1007/978-1-4612-1033-7
Background -- Differential Equations -- Differentiable Manifolds -- Partitions of Unity, Densities and Curves -- Critical Points -- Differential Forms -- Integration of Differential Forms -- Degree Theory -- Curves: The Local Theory -- Plane Curves: The Global Theory -- A Guide to the Local Theory of Surfaces in R3 -- A Guide to the Global Theory of Surfaces -- Bibliography -- Index of Symbols and Notations -- Index
This book is an introduction to modern differential geometry. The authors begin with the necessary tools from analysis and topology, including Sard's theorem, de Rham cohomology, calculus on manifolds, and a degree theory. The general theory is illustrated and expanded using the examples of curves and surfaces. In particular, the book contains the classical local and global theory of surfaces, including the fundamental forms, curvature, the Gauss-Bonnet formula, geodesics, and minimal surfaces