The Resource Differential Geometry : Connections, Curvature, and Characteristic Classes, by Loring W. Tu, (electronic resource)
Differential Geometry : Connections, Curvature, and Characteristic Classes, by Loring W. Tu, (electronic resource)
Resource Information
The item Differential Geometry : Connections, Curvature, and Characteristic Classes, by Loring W. Tu, (electronic resource) represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in European University Institute.This item is available to borrow from 1 library branch.
Resource Information
The item Differential Geometry : Connections, Curvature, and Characteristic Classes, by Loring W. Tu, (electronic resource) represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in European University Institute.
This item is available to borrow from 1 library branch.
 Summary
 This text presents a graduatelevel introduction to differential geometry for mathematics and physics students. The exposition follows the historical development of the concepts of connection and curvature with the goal of explaining the Chern–Weil theory of characteristic classes on a principal bundle. Along the way we encounter some of the high points in the history of differential geometry, for example, Gauss' Theorema Egregium and the Gauss–Bonnet theorem. Exercises throughout the book test the reader’s understanding of the material and sometimes illustrate extensions of the theory. Initially, the prerequisites for the reader include a passing familiarity with manifolds. After the first chapter, it becomes necessary to understand and manipulate differential forms. A knowledge of de Rham cohomology is required for the last third of the text. Prerequisite material is contained in author's text An Introduction to Manifolds, and can be learned in one semester. For the benefit of the reader and to establish common notations, Appendix A recalls the basics of manifold theory. Additionally, in an attempt to make the exposition more selfcontained, sections on algebraic constructions such as the tensor product and the exterior power are included. Differential geometry, as its name implies, is the study of geometry using differential calculus. It dates back to Newton and Leibniz in the seventeenth century, but it was not until the nineteenth century, with the work of Gauss on surfaces and Riemann on the curvature tensor, that differential geometry flourished and its modern foundation was laid. Over the past one hundred years, differential geometry has proven indispensable to an understanding of the physical world, in Einstein's general theory of relativity, in the theory of gravitation, in gauge theory, and now in string theory. Differential geometry is also useful in topology, several complex variables, algebraic geometry, complex manifolds, and dynamical systems, among other fields. The field has even found applications to group theory as in Gromov's work and to probability theory as in Diaconis's work. It is not too farfetched to argue that differential geometry should be in every mathematician's arsenal.
 Language
 eng
 Extent
 1 online resource (XVII, 347 pages)
 Contents

 Preface
 Chapter 1. Curvature and Vector Fields
 1. Riemannian Manifolds
 2. Curves
 3. Surfaces in Space
 4. Directional Derivative in Euclidean Space
 5. The Shape Operator
 6. Affine Connections
 7. Vector Bundles
 8. Gauss's Theorema Egregium
 9. Generalizations to Hypersurfaces in Rn+1
 Chapter 2. Curvature and Differential Forms
 10. Connections on a Vector Bundle
 11. Connection, Curvature, and Torsion Forms
 12. The Theorema Egregium Using Forms
 Chapter 3. Geodesics
 13. More on Affine Connections
 14. Geodesics
 15. Exponential Maps
 16. Distance and Volume
 17. The GaussBonnet Theorem
 Chapter 4. Tools from Algebra and Topology
 18. The Tensor Product and the Dual Module
 19. The Exterior Power
 20. Operations on Vector Bundles
 21. VectorValued Forms
 Chapter 5. Vector Bundles and Characteristic Classes
 22. Connections and Curvature Again
 23. Characteristic Classes
 24. Pontrjagin Classes
 25. The Euler Class and Chern Classes
 26. Some Applications of Characteristic Classes
 Chapter 6. Principal Bundles and Characteristic Classes
 27. Principal Bundles
 28. Connections on a Principal Bundle
 29. Horizontal Distributions on a Frame Bundle
 30. Curvature on a Principal Bundle
 31. Covariant Derivative on a Principal Bundle
 32. Character Classes of Principal Bundles
 A. Manifolds
 B. Invariant Polynomials
 Hints and Solutions to Selected EndofSection Problems
 List of Notations
 References
 Index
 Isbn
 9783319550848
 Label
 Differential Geometry : Connections, Curvature, and Characteristic Classes
 Title
 Differential Geometry
 Title remainder
 Connections, Curvature, and Characteristic Classes
 Statement of responsibility
 by Loring W. Tu
 Language
 eng
 Summary
 This text presents a graduatelevel introduction to differential geometry for mathematics and physics students. The exposition follows the historical development of the concepts of connection and curvature with the goal of explaining the Chern–Weil theory of characteristic classes on a principal bundle. Along the way we encounter some of the high points in the history of differential geometry, for example, Gauss' Theorema Egregium and the Gauss–Bonnet theorem. Exercises throughout the book test the reader’s understanding of the material and sometimes illustrate extensions of the theory. Initially, the prerequisites for the reader include a passing familiarity with manifolds. After the first chapter, it becomes necessary to understand and manipulate differential forms. A knowledge of de Rham cohomology is required for the last third of the text. Prerequisite material is contained in author's text An Introduction to Manifolds, and can be learned in one semester. For the benefit of the reader and to establish common notations, Appendix A recalls the basics of manifold theory. Additionally, in an attempt to make the exposition more selfcontained, sections on algebraic constructions such as the tensor product and the exterior power are included. Differential geometry, as its name implies, is the study of geometry using differential calculus. It dates back to Newton and Leibniz in the seventeenth century, but it was not until the nineteenth century, with the work of Gauss on surfaces and Riemann on the curvature tensor, that differential geometry flourished and its modern foundation was laid. Over the past one hundred years, differential geometry has proven indispensable to an understanding of the physical world, in Einstein's general theory of relativity, in the theory of gravitation, in gauge theory, and now in string theory. Differential geometry is also useful in topology, several complex variables, algebraic geometry, complex manifolds, and dynamical systems, among other fields. The field has even found applications to group theory as in Gromov's work and to probability theory as in Diaconis's work. It is not too farfetched to argue that differential geometry should be in every mathematician's arsenal.
 Assigning source
 Provided by publisher
 http://library.link/vocab/creatorName
 Tu, Loring W
 Image bit depth
 0
 Literary form
 non fiction
 Nature of contents
 dictionaries
 Series statement

 Springer eBooks
 Graduate Texts in Mathematics,
 Series volume
 275
 http://library.link/vocab/subjectName

 Mathematics
 Algebraic geometry
 Differential geometry
 Label
 Differential Geometry : Connections, Curvature, and Characteristic Classes, by Loring W. Tu, (electronic resource)
 Antecedent source
 mixed
 Carrier category
 online resource
 Carrier category code

 cr
 Carrier MARC source
 rdacarrier
 Color
 not applicable
 Content category
 text
 Content type code

 txt
 Content type MARC source
 rdacontent
 Contents
 Preface  Chapter 1. Curvature and Vector Fields  1. Riemannian Manifolds  2. Curves  3. Surfaces in Space  4. Directional Derivative in Euclidean Space  5. The Shape Operator  6. Affine Connections  7. Vector Bundles  8. Gauss's Theorema Egregium  9. Generalizations to Hypersurfaces in Rn+1  Chapter 2. Curvature and Differential Forms  10. Connections on a Vector Bundle  11. Connection, Curvature, and Torsion Forms  12. The Theorema Egregium Using Forms  Chapter 3. Geodesics  13. More on Affine Connections  14. Geodesics  15. Exponential Maps  16. Distance and Volume  17. The GaussBonnet Theorem  Chapter 4. Tools from Algebra and Topology  18. The Tensor Product and the Dual Module  19. The Exterior Power  20. Operations on Vector Bundles  21. VectorValued Forms  Chapter 5. Vector Bundles and Characteristic Classes  22. Connections and Curvature Again  23. Characteristic Classes  24. Pontrjagin Classes  25. The Euler Class and Chern Classes  26. Some Applications of Characteristic Classes  Chapter 6. Principal Bundles and Characteristic Classes  27. Principal Bundles  28. Connections on a Principal Bundle  29. Horizontal Distributions on a Frame Bundle  30. Curvature on a Principal Bundle  31. Covariant Derivative on a Principal Bundle  32. Character Classes of Principal Bundles  A. Manifolds  B. Invariant Polynomials  Hints and Solutions to Selected EndofSection Problems  List of Notations  References  Index
 Control code
 9783319550848
 Dimensions
 unknown
 Extent
 1 online resource (XVII, 347 pages)
 File format
 multiple file formats
 Form of item

 online
 electronic
 Governing access note
 Use of this electronic resource may be governed by a license agreement which restricts use to the European University Institute community. Each user is responsible for limiting use to individual, noncommercial purposes, without systematically downloading, distributing, or retaining substantial portions of information, provided that all copyright and other proprietary notices contained on the materials are retained. The use of software, including scripts, agents, or robots, is generally prohibited and may result in the loss of access to these resources for the entire European University Institute community
 Isbn
 9783319550848
 Level of compression
 uncompressed
 Media category
 computer
 Media MARC source
 rdamedia
 Media type code

 c
 Other control number
 10.1007/9783319550848
 Other physical details
 87 illustrations, 15 illustrations in color.
 Quality assurance targets
 absent
 Reformatting quality
 access
 Specific material designation
 remote
 System control number
 (OCoLC)989727720
 Label
 Differential Geometry : Connections, Curvature, and Characteristic Classes, by Loring W. Tu, (electronic resource)
 Antecedent source
 mixed
 Carrier category
 online resource
 Carrier category code

 cr
 Carrier MARC source
 rdacarrier
 Color
 not applicable
 Content category
 text
 Content type code

 txt
 Content type MARC source
 rdacontent
 Contents
 Preface  Chapter 1. Curvature and Vector Fields  1. Riemannian Manifolds  2. Curves  3. Surfaces in Space  4. Directional Derivative in Euclidean Space  5. The Shape Operator  6. Affine Connections  7. Vector Bundles  8. Gauss's Theorema Egregium  9. Generalizations to Hypersurfaces in Rn+1  Chapter 2. Curvature and Differential Forms  10. Connections on a Vector Bundle  11. Connection, Curvature, and Torsion Forms  12. The Theorema Egregium Using Forms  Chapter 3. Geodesics  13. More on Affine Connections  14. Geodesics  15. Exponential Maps  16. Distance and Volume  17. The GaussBonnet Theorem  Chapter 4. Tools from Algebra and Topology  18. The Tensor Product and the Dual Module  19. The Exterior Power  20. Operations on Vector Bundles  21. VectorValued Forms  Chapter 5. Vector Bundles and Characteristic Classes  22. Connections and Curvature Again  23. Characteristic Classes  24. Pontrjagin Classes  25. The Euler Class and Chern Classes  26. Some Applications of Characteristic Classes  Chapter 6. Principal Bundles and Characteristic Classes  27. Principal Bundles  28. Connections on a Principal Bundle  29. Horizontal Distributions on a Frame Bundle  30. Curvature on a Principal Bundle  31. Covariant Derivative on a Principal Bundle  32. Character Classes of Principal Bundles  A. Manifolds  B. Invariant Polynomials  Hints and Solutions to Selected EndofSection Problems  List of Notations  References  Index
 Control code
 9783319550848
 Dimensions
 unknown
 Extent
 1 online resource (XVII, 347 pages)
 File format
 multiple file formats
 Form of item

 online
 electronic
 Governing access note
 Use of this electronic resource may be governed by a license agreement which restricts use to the European University Institute community. Each user is responsible for limiting use to individual, noncommercial purposes, without systematically downloading, distributing, or retaining substantial portions of information, provided that all copyright and other proprietary notices contained on the materials are retained. The use of software, including scripts, agents, or robots, is generally prohibited and may result in the loss of access to these resources for the entire European University Institute community
 Isbn
 9783319550848
 Level of compression
 uncompressed
 Media category
 computer
 Media MARC source
 rdamedia
 Media type code

 c
 Other control number
 10.1007/9783319550848
 Other physical details
 87 illustrations, 15 illustrations in color.
 Quality assurance targets
 absent
 Reformatting quality
 access
 Specific material designation
 remote
 System control number
 (OCoLC)989727720
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<div class="citation" vocab="http://schema.org/"><i class="fa faexternallinksquare fafw"></i> Data from <span resource="http://link.library.eui.eu/portal/DifferentialGeometryConnectionsCurvature/pm76OrcWjks/" typeof="Book http://bibfra.me/vocab/lite/Item"><span property="name http://bibfra.me/vocab/lite/label"><a href="http://link.library.eui.eu/portal/DifferentialGeometryConnectionsCurvature/pm76OrcWjks/">Differential Geometry : Connections, Curvature, and Characteristic Classes, by Loring W. Tu, (electronic resource)</a></span>  <span property="potentialAction" typeOf="OrganizeAction"><span property="agent" typeof="LibrarySystem http://library.link/vocab/LibrarySystem" resource="http://link.library.eui.eu/"><span property="name http://bibfra.me/vocab/lite/label"><a property="url" href="http://link.library.eui.eu/">European University Institute</a></span></span></span></span></div>