Isometric Embeddings of Riemannian and Pseudo-Riemannian Manifolds
| Author | : Robert Everist Greene |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 69 |
| Release | : 1970 |
| Genre | : Embeddings (Mathematics) |
| ISBN | : 0821812971 |
Isometric Embeddings Of Riemannian And Pseudo Riemannian Manifolds
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Isometric Embedding of Riemannian Manifolds in Euclidean Spaces
| Author | : Qing Han |
| Publisher | : American Mathematical Soc. |
| Total 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.
Pseudo-Riemannian Geometry, [delta]-invariants and Applications
| Author | : Bang-yen Chen |
| Publisher | : World Scientific |
| Total Pages | : 510 |
| Release | : 2011 |
| Genre | : Mathematics |
| ISBN | : 9814329649 |
The first part of this book provides a self-contained and accessible introduction to the subject in the general setting of pseudo-Riemannian manifolds and their non-degenerate submanifolds, only assuming from the reader some basic knowledge about manifold
Introduction to Riemannian Manifolds
| Author | : John M. Lee |
| Publisher | : Springer |
| Total Pages | : 447 |
| Release | : 2019-01-02 |
| Genre | : Mathematics |
| ISBN | : 3319917552 |
This text focuses on developing an intimate acquaintance with the geometric meaning of curvature and thereby introduces and demonstrates all the main technical tools needed for a more advanced course on Riemannian manifolds. It covers proving the four most fundamental theorems relating curvature and topology: the Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnet’s Theorem, and a special case of the Cartan-Ambrose-Hicks Theorem.
Differential Geometry: Riemannian Geometry
| Author | : Robert Everist Greene |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 735 |
| Release | : 1993 |
| Genre | : Mathematics |
| ISBN | : 0821814966 |
The third of three parts comprising Volume 54, the proceedings of the Summer Research Institute on Differential Geometry, held at the University of California, Los Angeles, July 1990 (ISBN for the set is 0-8218-1493-1). Part 3 begins with an overview by R.E. Greene of some recent trends in Riemannia
An Introduction to Riemannian Geometry
| Author | : Leonor Godinho |
| Publisher | : Springer |
| Total Pages | : 476 |
| Release | : 2014-07-26 |
| Genre | : Mathematics |
| ISBN | : 3319086669 |
Unlike many other texts on differential geometry, this textbook also offers interesting applications to geometric mechanics and general relativity. The first part is a concise and self-contained introduction to the basics of manifolds, differential forms, metrics and curvature. The second part studies applications to mechanics and relativity including the proofs of the Hawking and Penrose singularity theorems. It can be independently used for one-semester courses in either of these subjects. The main ideas are illustrated and further developed by numerous examples and over 300 exercises. Detailed solutions are provided for many of these exercises, making An Introduction to Riemannian Geometry ideal for self-study.
Differential Geometry and Continuum Mechanics
| Author | : Gui-Qiang G. Chen |
| Publisher | : Springer |
| Total Pages | : 384 |
| Release | : 2015-08-11 |
| Genre | : Mathematics |
| ISBN | : 331918573X |
This book examines the exciting interface between differential geometry and continuum mechanics, now recognised as being of increasing technological significance. Topics discussed include isometric embeddings in differential geometry and the relation with microstructure in nonlinear elasticity, the use of manifolds in the description of microstructure in continuum mechanics, experimental measurement of microstructure, defects, dislocations, surface energies, and nematic liquid crystals. Compensated compactness in partial differential equations is also treated. The volume is intended for specialists and non-specialists in pure and applied geometry, continuum mechanics, theoretical physics, materials and engineering sciences, and partial differential equations. It will also be of interest to postdoctoral scientists and advanced postgraduate research students. These proceedings include revised written versions of the majority of papers presented by leading experts at the ICMS Edinburgh Workshop on Differential Geometry and Continuum Mechanics held in June 2013. All papers have been peer reviewed.
Differential Systems and Isometric Embeddings.(AM-114), Volume 114
| Author | : Phillip A. Griffiths |
| Publisher | : Princeton University Press |
| Total Pages | : 238 |
| Release | : 2016-03-02 |
| Genre | : Mathematics |
| ISBN | : 1400882109 |
The theory of exterior differential systems provides a framework for systematically addressing the typically non-linear, and frequently overdetermined, partial differential equations that arise in differential geometry. Adaptation of the techniques of microlocalization to differential systems have led to recent activity on the foundations of the theory; in particular, the fundamental role of the characteristic variety in geometric problems is now clearly established. In this book the general theory is explained in a relatively quick and concrete manner, and then this general theory is applied to the recent developments in the classical problem of isometric embeddings of Riemannian manifolds.
Seminar on Differential Geometry. (AM-102), Volume 102
| Author | : Shing-tung Yau |
| Publisher | : Princeton University Press |
| Total Pages | : 720 |
| Release | : 2016-03-02 |
| Genre | : Mathematics |
| ISBN | : 1400881919 |
This collection of papers constitutes a wide-ranging survey of recent developments in differential geometry and its interactions with other fields, especially partial differential equations and mathematical physics. This area of mathematics was the subject of a special program at the Institute for Advanced Study in Princeton during the academic year 1979-1980; the papers in this volume were contributed by the speakers in the sequence of seminars organized by Shing-Tung Yau for this program. Both survey articles and articles presenting new results are included. The articles on differential geometry and partial differential equations include a general survey article by the editor on the relationship of the two fields and more specialized articles on topics including harmonic mappings, isoperimetric and Poincaré inequalities, metrics with specified curvature properties, the Monge-Arnpere equation, L2 harmonic forms and cohomology, manifolds of positive curvature, isometric embedding, and Kraumlhler manifolds and metrics. The articles on differential geometry and mathematical physics cover such topics as renormalization, instantons, gauge fields and the Yang-Mills equation, nonlinear evolution equations, incompleteness of space-times, black holes, and quantum gravity. A feature of special interest is the inclusion of a list of more than one hundred unsolved research problems compiled by the editor with comments and bibliographical information.
Explorations in Complex and Riemannian Geometry
| Author | : John Bland |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 338 |
| Release | : 2003 |
| Genre | : Mathematics |
| ISBN | : 0821832735 |
This book contains contributions by an impressive list of leading mathematicians. The articles include high-level survey and research papers exploring contemporary issues in geometric analysis, differential geometry, and several complex variables. Many of the articles will provide graduate students with a good entry point into important areas of modern research. The material is intended for researchers and graduate students interested in several complex variables and complex geometry.
