The Resource PseudoRiemannian geometry, [delta]invariants and applications, BangYen Chen
PseudoRiemannian geometry, [delta]invariants and applications, BangYen Chen
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The item PseudoRiemannian geometry, [delta]invariants and applications, BangYen Chen represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in University of MissouriSt. Louis Libraries.This item is available to borrow from 1 library branch.
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The item PseudoRiemannian geometry, [delta]invariants and applications, BangYen Chen represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in University of MissouriSt. Louis Libraries.
This item is available to borrow from 1 library branch.
 Summary
 The first part of this book provides a selfcontained and accessible introduction to the subject in the general setting of pseudoRiemannian manifolds and their nondegenerate submanifolds, only assuming from the reader some basic knowledge about manifold theory. A number of recent results on pseudoRiemannian submanifolds are also included. The second part of this book is on [symbol]invariants, which was introduced in the early 1990s by the author. The famous Nash embedding theorem published in 1956 was aimed for, in the hope that if Riemannian manifolds could be regarded as Riemannian submanifolds, this would then yield the opportunity to use extrinsic help. However, this hope had not been materialized as pointed out by M. Gromov in his 1985 article published in Asterisque. The main reason for this is the lack of control of the extrinsic invariants of the submanifolds by known intrinsic invariants. In order to overcome such difficulties, as well as to provide answers for an open question on minimal immersions, the author introduced in the early 1990s new types of Riemannian invariants, known as [symbol]invariants, which are very different in nature from the classical Ricci and scalar curvatures. At the same time he was able to establish general optimal relations between [symbol]invariants and the main extrinsic invariants. Since then many new results concerning these [symbol]invariants have been obtained by many geometers. The second part of this book is to provide an extensive and comprehensive survey over this very active field of research done during the last two decades
 Language
 eng
 Extent
 1 online resource (xxxii, 477 pages)
 Contents

 1. PseudoRiemannian manifolds. 1.1. Symmetric bilinear forms and scalar products. 1.2. PseudoRiemannian manifolds. 1.3. Physical interpretations of pseudoRiemannian manifolds. 1.4. LeviCivita connection. 1.5. Parallel translation. 1.6. Riemann curvature tensor. 1.7. Sectional, Ricci and scalar curvatures. 1.8. Indefinite real space forms. 1.9. Lie derivative, gradient, Hessian and Laplacian. 1.10. Weyl conformal curvature tensor
 2. Basics on pseudoRiemannian submanifolds. 2.1. Isometric immersions. 2.2. CartanJanet's and Nash's embedding theorems. 2.3. Gauss' formula and second fundamental form. 2.4. Weingarten's formula and normal connection. 2.5. Shape operator of pseudoRiemannian submanifolds. 2.6. Fundamental equations of Gauss, Codazzi and Ricci. 2.7. Fundamental theorems of submanifolds. 2.8. A reduction theorem of ErbacherMagid. 2.9. Two basic formulas for submanifolds in E[symbol]. 2.10. Relationship between squared mean curvature and Ricci curvature. 2.11. Relationship between shape operator and Ricci curvature. 2.12. Cartan's structure equations
 3. Special pseudoRiemannian submanifolds. 3.1. Totally geodesic submanifolds. 3.2. Parallel submanifolds of (indefinite) real space forms. 3.3. Totally umbilical submanifolds. 3.4. Totally umbilical submanifolds of S[symbol] (1) and H[symbol] ( 1). 3.5. Pseudoumbilical submanifolds of E[symbol]. 3.6. Pseudoumbilical submanifolds of S[symbol] (1) and H[symbol] ( 1). 3.7. Minimal Lorentz surfaces in indefinite real space forms. 3.8. Marginally trapped surfaces and black holes. 3.9. Quasiminimal surfaces in indefinite space forms
 4. Warped products and twisted products. 4.1. Basics of warped products. 4.2. Curvature of warped products. 4.3. Warped product immersions. 4.4. Twisted products. 4.5. Doubletwisted products and their characterization
 5. RobertsonWalker spacetimes. 5.1. Cosmology, RobertsonWalker spacetimes and Einstein's field equations. 5.2. Basic properties of RobertsonWalker spacetimes. 5.3. Totally geodesic submanifolds of RW spacetimes. 5.4. Parallel submanifolds of RW spacetimes. 5.5. Totally umbilical submanifolds of RW spacetimes. 5.6. Hypersurfaces of constant curvature in RW spacetimes. 5.7. Realization of RW spacetimes in pseudoEuclidean spaces
 Isbn
 9789814329644
 Label
 PseudoRiemannian geometry, [delta]invariants and applications
 Title
 PseudoRiemannian geometry, [delta]invariants and applications
 Statement of responsibility
 BangYen Chen
 Title variation
 PseudoRiemannian geometry, dinvariants and applications
 Language
 eng
 Summary
 The first part of this book provides a selfcontained and accessible introduction to the subject in the general setting of pseudoRiemannian manifolds and their nondegenerate submanifolds, only assuming from the reader some basic knowledge about manifold theory. A number of recent results on pseudoRiemannian submanifolds are also included. The second part of this book is on [symbol]invariants, which was introduced in the early 1990s by the author. The famous Nash embedding theorem published in 1956 was aimed for, in the hope that if Riemannian manifolds could be regarded as Riemannian submanifolds, this would then yield the opportunity to use extrinsic help. However, this hope had not been materialized as pointed out by M. Gromov in his 1985 article published in Asterisque. The main reason for this is the lack of control of the extrinsic invariants of the submanifolds by known intrinsic invariants. In order to overcome such difficulties, as well as to provide answers for an open question on minimal immersions, the author introduced in the early 1990s new types of Riemannian invariants, known as [symbol]invariants, which are very different in nature from the classical Ricci and scalar curvatures. At the same time he was able to establish general optimal relations between [symbol]invariants and the main extrinsic invariants. Since then many new results concerning these [symbol]invariants have been obtained by many geometers. The second part of this book is to provide an extensive and comprehensive survey over this very active field of research done during the last two decades
 Cataloging source
 N$T
 http://library.link/vocab/creatorName
 Chen, Bangyen
 Dewey number
 516.36
 Illustrations
 illustrations
 Index
 index present
 LC call number
 QA649
 LC item number
 .C482 2011eb
 Literary form
 non fiction
 Nature of contents

 dictionaries
 bibliography
 http://library.link/vocab/subjectName

 Submanifolds
 Riemannian manifolds
 Geometry, Riemannian
 Invariants
 MATHEMATICS
 Geometry, Riemannian
 Invariants
 Riemannian manifolds
 Submanifolds
 Label
 PseudoRiemannian geometry, [delta]invariants and applications, BangYen Chen
 Antecedent source
 unknown
 Bibliography note
 Includes bibliographical references (pages 439462) and indexes
 Carrier category
 online resource
 Carrier category code

 cr
 Carrier MARC source
 rdacarrier
 Color
 multicolored
 Content category
 text
 Content type code

 txt
 Content type MARC source
 rdacontent
 Contents
 1. PseudoRiemannian manifolds. 1.1. Symmetric bilinear forms and scalar products. 1.2. PseudoRiemannian manifolds. 1.3. Physical interpretations of pseudoRiemannian manifolds. 1.4. LeviCivita connection. 1.5. Parallel translation. 1.6. Riemann curvature tensor. 1.7. Sectional, Ricci and scalar curvatures. 1.8. Indefinite real space forms. 1.9. Lie derivative, gradient, Hessian and Laplacian. 1.10. Weyl conformal curvature tensor  2. Basics on pseudoRiemannian submanifolds. 2.1. Isometric immersions. 2.2. CartanJanet's and Nash's embedding theorems. 2.3. Gauss' formula and second fundamental form. 2.4. Weingarten's formula and normal connection. 2.5. Shape operator of pseudoRiemannian submanifolds. 2.6. Fundamental equations of Gauss, Codazzi and Ricci. 2.7. Fundamental theorems of submanifolds. 2.8. A reduction theorem of ErbacherMagid. 2.9. Two basic formulas for submanifolds in E[symbol]. 2.10. Relationship between squared mean curvature and Ricci curvature. 2.11. Relationship between shape operator and Ricci curvature. 2.12. Cartan's structure equations  3. Special pseudoRiemannian submanifolds. 3.1. Totally geodesic submanifolds. 3.2. Parallel submanifolds of (indefinite) real space forms. 3.3. Totally umbilical submanifolds. 3.4. Totally umbilical submanifolds of S[symbol] (1) and H[symbol] ( 1). 3.5. Pseudoumbilical submanifolds of E[symbol]. 3.6. Pseudoumbilical submanifolds of S[symbol] (1) and H[symbol] ( 1). 3.7. Minimal Lorentz surfaces in indefinite real space forms. 3.8. Marginally trapped surfaces and black holes. 3.9. Quasiminimal surfaces in indefinite space forms  4. Warped products and twisted products. 4.1. Basics of warped products. 4.2. Curvature of warped products. 4.3. Warped product immersions. 4.4. Twisted products. 4.5. Doubletwisted products and their characterization  5. RobertsonWalker spacetimes. 5.1. Cosmology, RobertsonWalker spacetimes and Einstein's field equations. 5.2. Basic properties of RobertsonWalker spacetimes. 5.3. Totally geodesic submanifolds of RW spacetimes. 5.4. Parallel submanifolds of RW spacetimes. 5.5. Totally umbilical submanifolds of RW spacetimes. 5.6. Hypersurfaces of constant curvature in RW spacetimes. 5.7. Realization of RW spacetimes in pseudoEuclidean spaces
 Control code
 754793931
 Dimensions
 unknown
 Extent
 1 online resource (xxxii, 477 pages)
 File format
 unknown
 Form of item
 online
 Isbn
 9789814329644
 Level of compression
 unknown
 Media category
 computer
 Media MARC source
 rdamedia
 Media type code

 c
 Other physical details
 illustrations
 Quality assurance targets
 not applicable
 Reformatting quality
 unknown
 Sound
 unknown sound
 Specific material designation
 remote
 System control number
 (OCoLC)754793931
 Label
 PseudoRiemannian geometry, [delta]invariants and applications, BangYen Chen
 Antecedent source
 unknown
 Bibliography note
 Includes bibliographical references (pages 439462) and indexes
 Carrier category
 online resource
 Carrier category code

 cr
 Carrier MARC source
 rdacarrier
 Color
 multicolored
 Content category
 text
 Content type code

 txt
 Content type MARC source
 rdacontent
 Contents
 1. PseudoRiemannian manifolds. 1.1. Symmetric bilinear forms and scalar products. 1.2. PseudoRiemannian manifolds. 1.3. Physical interpretations of pseudoRiemannian manifolds. 1.4. LeviCivita connection. 1.5. Parallel translation. 1.6. Riemann curvature tensor. 1.7. Sectional, Ricci and scalar curvatures. 1.8. Indefinite real space forms. 1.9. Lie derivative, gradient, Hessian and Laplacian. 1.10. Weyl conformal curvature tensor  2. Basics on pseudoRiemannian submanifolds. 2.1. Isometric immersions. 2.2. CartanJanet's and Nash's embedding theorems. 2.3. Gauss' formula and second fundamental form. 2.4. Weingarten's formula and normal connection. 2.5. Shape operator of pseudoRiemannian submanifolds. 2.6. Fundamental equations of Gauss, Codazzi and Ricci. 2.7. Fundamental theorems of submanifolds. 2.8. A reduction theorem of ErbacherMagid. 2.9. Two basic formulas for submanifolds in E[symbol]. 2.10. Relationship between squared mean curvature and Ricci curvature. 2.11. Relationship between shape operator and Ricci curvature. 2.12. Cartan's structure equations  3. Special pseudoRiemannian submanifolds. 3.1. Totally geodesic submanifolds. 3.2. Parallel submanifolds of (indefinite) real space forms. 3.3. Totally umbilical submanifolds. 3.4. Totally umbilical submanifolds of S[symbol] (1) and H[symbol] ( 1). 3.5. Pseudoumbilical submanifolds of E[symbol]. 3.6. Pseudoumbilical submanifolds of S[symbol] (1) and H[symbol] ( 1). 3.7. Minimal Lorentz surfaces in indefinite real space forms. 3.8. Marginally trapped surfaces and black holes. 3.9. Quasiminimal surfaces in indefinite space forms  4. Warped products and twisted products. 4.1. Basics of warped products. 4.2. Curvature of warped products. 4.3. Warped product immersions. 4.4. Twisted products. 4.5. Doubletwisted products and their characterization  5. RobertsonWalker spacetimes. 5.1. Cosmology, RobertsonWalker spacetimes and Einstein's field equations. 5.2. Basic properties of RobertsonWalker spacetimes. 5.3. Totally geodesic submanifolds of RW spacetimes. 5.4. Parallel submanifolds of RW spacetimes. 5.5. Totally umbilical submanifolds of RW spacetimes. 5.6. Hypersurfaces of constant curvature in RW spacetimes. 5.7. Realization of RW spacetimes in pseudoEuclidean spaces
 Control code
 754793931
 Dimensions
 unknown
 Extent
 1 online resource (xxxii, 477 pages)
 File format
 unknown
 Form of item
 online
 Isbn
 9789814329644
 Level of compression
 unknown
 Media category
 computer
 Media MARC source
 rdamedia
 Media type code

 c
 Other physical details
 illustrations
 Quality assurance targets
 not applicable
 Reformatting quality
 unknown
 Sound
 unknown sound
 Specific material designation
 remote
 System control number
 (OCoLC)754793931
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