Isometric embedding of Riemannian manifolds in Euclidean spaces / Qing Han, Jia-Xing Hong.

Format:

Book

Author/Creator:

Han, Qing, author.

Hong, Jia-Xing, 1942- author.

Series:

Mathematical surveys and monographs ; volume 130.

Mathematical surveys and monographs ; volume 130

Language:

English

Subjects (All):

Riemannian manifolds.

Isometrics (Mathematics).

Algebraic spaces.

Physical Description:

1 online resource (278 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2006]

Language Note:

English

Summary:

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.

Contents:

""Contents""; ""Preface""; ""A Brief History""; ""Part 1. Isometric Embedding of Riemannian Manifolds""; ""Chapter 1. Fundamental Theorems""; ""1.1. Local Isometric Embedding of Analytic Metrics""; ""1.2. Local Isometric Embedding of Smooth Metrics""; ""1.3. Global Isometric Embedding of Smooth Metrics""; ""Notes""; ""Chapter 2. Surfaces in Low Dimensional Euclidean Spaces""; ""2.1. Surfaces in R[sup(3)]""; ""2.2. Isometric Immersions of Surfaces of Constant Curvature""; ""2.3. Isometric Immersions of Surfaces in M[sup(4)]""; ""Notes""

""Part 2. Local Isometric Embedding of Surfaces in R[sup(3)]""""Chapter 3. Basic Equations""; ""3.1. The Darboux Equation""; ""3.2. The Gauss-Codazzi System""; ""3.3. The Rozhdestvenskii-Poznyak System""; ""Notes""; ""Chapter 4. Nonzero Gauss Curvature""; ""4.1. Positive Gauss Curvature""; ""4.2. Negative Gauss Curvature""; ""4.3. Appendix: Sobolev Spaces""; ""4.4. Appendix: Some Lemmas""; ""4.5. Appendix: Symmetric Hyperbolic Linear Differential Systems""; ""Notes""; ""Chapter 5. Gauss Curvature Changing Sign Cleanly""; ""5.1. The Setting""; ""5.2. Iterations""

""5.3. Symmetric Positive Linear Differential Systems""""Notes""; ""Chapter 6. Nonnegative Gauss Curvature""; ""6.1. The Setting""; ""6.2. A Priori Estimates""; ""6.3. Iterations""; ""Notes""; ""Chapter 7. Nonpositive Gauss Curvature""; ""7.1. The Setting""; ""7.2. A Priori Estimates for the Linearized System""; ""7.3. A Priori Estimates for the Cauchy Problem""; ""7.4. Iterations""; ""7.5. Remarks on the Hypothesis""; ""Notes""; ""Part 3. Global Isometric Embedding of Surfaces in R[sup(3)]""; ""Chapter 8. Deformation of Surfaces""; ""8.1. The Rigidity of Surfaces""

""8.2. The Infinitesimal Rigidity of Surfaces""""8.3. A Positive Disk of Not Infinitesimal Rigidity""; ""Notes""; ""Chapter 9. The Weyl Problem""; ""9.1. Geometric Preliminaries""; ""9.2. Openness""; ""9.3. Closedness""; ""Notes""; ""Chapter 10. Complete Negatively Curved Surfaces""; ""10.1. Nonexistence of Isometric Immersions""; ""10.2. Existence of Isometric Immersions, Part I""; ""10.3. Existence of Isometric Immersions, Part II""; ""Notes""; ""Chapter 11. Boundary Value Problems""; ""11.1. The Dirichlet Problem with Planar Boundaries""

""11.2. The Darboux System and Interior C[sup(2)]-Estimates""""11.3. Global C[sup(2)]-Estimates""; ""Notes""; ""Bibliography""; ""Index""; ""A""; ""C""; ""D""; ""E""; ""F""; ""G""; ""H""; ""I""; ""J""; ""K""; ""L""; ""M""; ""N""; ""O""; ""P""; ""R""; ""S""; ""T""; ""U""; ""W""; ""Y""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 249-257) and index.

Description based on print version record.

ISBN:

1-4704-1357-4