On Embedding Differentiable Manifolds in Euclidian Space
Title | On Embedding Differentiable Manifolds in Euclidian Space PDF eBook |
Author | Morris W. Hirsch |
Publisher | |
Pages | 20 |
Release | 1960 |
Genre | Manifolds (Mathematics) |
ISBN |
Isometric Embedding of Riemannian Manifolds in Euclidean Spaces
Title | Isometric Embedding of Riemannian Manifolds in Euclidean Spaces PDF eBook |
Author | Qing Han |
Publisher | American Mathematical Soc. |
Pages | 278 |
Release | 2006 |
Genre | Mathematics |
ISBN | 0821840711 |
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.
An Introduction To Differential Manifolds
Title | An Introduction To Differential Manifolds PDF eBook |
Author | Dennis Barden |
Publisher | World Scientific |
Pages | 231 |
Release | 2003-03-12 |
Genre | Mathematics |
ISBN | 1911298232 |
This invaluable book, based on the many years of teaching experience of both authors, introduces the reader to the basic ideas in differential topology. Among the topics covered are smooth manifolds and maps, the structure of the tangent bundle and its associates, the calculation of real cohomology groups using differential forms (de Rham theory), and applications such as the Poincaré-Hopf theorem relating the Euler number of a manifold and the index of a vector field. Each chapter contains exercises of varying difficulty for which solutions are provided. Special features include examples drawn from geometric manifolds in dimension 3 and Brieskorn varieties in dimensions 5 and 7, as well as detailed calculations for the cohomology groups of spheres and tori.
Manifolds And Local Structures: A General Theory
Title | Manifolds And Local Structures: A General Theory PDF eBook |
Author | Marco Grandis |
Publisher | World Scientific |
Pages | 374 |
Release | 2021-02-10 |
Genre | Mathematics |
ISBN | 9811234019 |
Local structures, like differentiable manifolds, fibre bundles, vector bundles and foliations, can be obtained by gluing together a family of suitable 'elementary spaces', by means of partial homeomorphisms that fix the gluing conditions and form a sort of 'intrinsic atlas', instead of the more usual system of charts living in an external framework.An 'intrinsic manifold' is defined here as such an atlas, in a suitable category of elementary spaces: open euclidean spaces, or trivial bundles, or trivial vector bundles, and so on.This uniform approach allows us to move from one basis to another: for instance, the elementary tangent bundle of an open Euclidean space is automatically extended to the tangent bundle of any differentiable manifold. The same holds for tensor calculus.Technically, the goal of this book is to treat these structures as 'symmetric enriched categories' over a suitable basis, generally an ordered category of partial mappings.This approach to gluing structures is related to Ehresmann's one, based on inductive pseudogroups and inductive categories. A second source was the theory of enriched categories and Lawvere's unusual view of interesting mathematical structures as categories enriched over a suitable basis.
On Embedding Differentiable Manifolds in Euclidian Space
Title | On Embedding Differentiable Manifolds in Euclidian Space PDF eBook |
Author | Morris W. Hirsch |
Publisher | |
Pages | 9 |
Release | 1960 |
Genre | Manifolds (Mathematics) |
ISBN |
Differentiable Manifolds
Title | Differentiable Manifolds PDF eBook |
Author | Sze-Tsen Hu |
Publisher | |
Pages | 196 |
Release | 1969 |
Genre | Mathematics |
ISBN |
An Introduction to Differentiable Manifolds and Riemannian Geometry
Title | An Introduction to Differentiable Manifolds and Riemannian Geometry PDF eBook |
Author | |
Publisher | Academic Press |
Pages | 447 |
Release | 1986-04-21 |
Genre | Mathematics |
ISBN | 0080874398 |
An Introduction to Differentiable Manifolds and Riemannian Geometry