Isometric Embedding of Riemannian Manifolds in Euclidean Spaces, Volume 13American Mathematical Soc., 2006 - 260 pages The question of the existence of isometric embeddings of Riemannian manifolds in Euclidean space is already more than a century old. This book presents, in a systematic way, results both local and global and in arbitrary dimension but with a focus on the isometric embedding of surfaces in ${\mathbb R}^3$. The emphasis is on those PDE techniques which are essential to the most important results of the last century. The classic results in this book include the Janet-Cartan Theorem, Nirenberg's solution of the Weyl problem, and Nash's Embedding Theorem, with a simplified proof by Gunther. The book also includes the main results from the past twenty years, both local and global, on the isometric embedding of surfaces in Euclidean 3-space. The work will be indispensable to researchers in the area. Moreover, the authors integrate the results and techniques into a unified whole, providing a good entry point into the area for advanced graduate students or anyone interested in this subject. The authors avoid what is technically complicated. Background knowledge is kept to an essential minimum: a one-semester course in differential geometry and a one-year course in partial differential equations. |
Table des matières
Part 2 Local Isometric Embedding of Surfaces in Rsup3 | 43 |
Part 3 Global Isometric Embedding of Surfaces in Rsup3 | 143 |
Notes | 247 |
249 | |
259 | |
Autres éditions - Tout afficher
Isometric Embedding of Riemannian Manifolds in Euclidean Spaces, Volume 13 Qing Han,Jia-Xing Hong,Jiaxing Hong Aucun aperçu disponible - 2006 |
Expressions et termes fréquents
analytic apply assume assumption asymptotic boundary bounded calculation chapter choose closed coefficients compact complete condition consider convex coordinates curves Darboux equation defined definite denote derivatives differential equations differential system discuss easy elliptic estimate exists expressions fact fixed function Gauss curvature geodesic given global Hence holds hyperbolic implies independent integral introduce isometric embedding isometric immersion iteration Lemma linear lines metric g Moreover negative neighborhood Note obtain Obviously operator origin plane positive constant problem proof proof of Theorem prove REMARK respect result Riemannian manifold rigid satisfies side similar smooth smooth function smooth isometric smooth solution solution solve space step Substituting sufficiently Suppose surface symmetric Theorem unique unit vector write yields zero