Differential Geometry Connections Curvature and Characteristic Classes Review
Differential Geometry : Connections, Curvature, and Characteristic Classes
Description
This text presents a graduate-level 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 primary package. 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 examination 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 sympathize and manipulate differential forms. A knowledge of de Rham cohomology is required for the last third of the text.Prerequisite material is independent in author's text An Introduction to Manifolds, and can be learned in ane semester. For the benefit of the reader and to plant common notations, Appendix A recalls the nuts of manifold theory. Additionally, in an attempt to brand the exposition more self-contained, sections on algebraic constructions such equally the tensor production and the exterior power are included.
Differential geometry, as its proper name implies, is the written report of geometry using differential calculus. Information technology dates dorsum 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 by 1 hundred years, differential geometry has proven indispensable to an understanding of the physical world, in Einstein's full general theory of relativity, in the theory of gravitation, in gauge theory, and now in cord theory. Differential geometry is as well useful in topology, several complex variables, algebraic geometry, complex manifolds, and dynamical systems, among other fields. The field has even plant applications to grouping theory as in Gromov'southward work and to probability theory equally in Diaconis's work. It is not too far-fetched to contend that differential geometry should be in every mathematician'south arsenal.
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Product details
- Hardback | 347 pages
- 155 ten 235 10 25.4mm | 7,409g
- 01 Jul 2017
- Springer International Publishing AG
- Cham, Switzerland
- English
- 1st ed. 2017
- 13 Tables, colour; 15 Illustrations, colour; 72 Illustrations, black and white; XVII, 347 p. 87 illus., fifteen illus. in color.
- 3319550829
- 9783319550824
- 621,983
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This text presents a graduate-level introduction to differential geometry for mathematics and physics students. The exposition follows the historical evolution of the concepts of connection and curvature with the goal of explaining the Chern-Weil theory of characteristic classes on a chief package. Forth the way nosotros 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 cognition of de Rham cohomology is required for the terminal tertiary of the text.
Prerequisite material is contained in writer'south text An Introduction to Manifolds, and can be learned in i 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 brand the exposition more self-contained, sections on algebraic constructions such as the tensor product and the exterior ability are included.
Differential geometry, equally its name implies, is the study of geometry using differential calculus. Information technology dates back to Newton and Leibniz in the seventeenth century, but it was not until the nineteenth century, with the piece of work of Gauss on surfaces and Riemann on the curvature tensor, that differential geometry flourished and its modern foundation was laid. Over the past i hundred years, differential geometry has proven indispensable to an agreement 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, circuitous manifolds, and dynamical systems, among other fields. The field has even found applications to group theory as in Gromov'southward piece of work and to probability theory every bit in Diaconis's work. It is non too far-fetched to argue that differential geometry should be in every mathematician'due south arsenal.
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Tabular array of contents
Preface.- Affiliate 1. Curvature and Vector Fields.- 1. Riemannian Manifolds.- ii. Curves.- 3. Surfaces in Space.- 4. Directional Derivative in Euclidean Space.- v. The Shape Operator.- 6. Affine Connections.- vii. Vector Bundles.- 8. Gauss's Theorema Egregium.- 9. Generalizations to Hypersurfaces in Rn+i.- Chapter 2. Curvature and Differential Forms.- 10. Connections on a Vector Parcel.- 11. Connexion, Curvature, and Torsion Forms.- 12. The Theorema Egregium Using Forms.- Affiliate iii. Geodesics.- thirteen. More on Affine Connections.- 14. Geodesics.- xv. Exponential Maps.- xvi. Distance and Book.- 17. The Gauss-Bonnet Theorem.- Chapter 4. Tools from Algebra and Topology.- 18. The Tensor Product and the Dual Module.- nineteen. The Exterior Power.- xx. Operations on Vector Bundles.- 21. Vector-Valued Forms.- Chapter v. 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. Master Bundles.- 28. Connections on a Principal Bundle.- 29. Horizontal Distributions on a Frame Bundle.- 30. Curvature on a Principal Package.- 31. Covariant Derivative on a Principal Packet.- 32. Character Classes of Principal Bundles.- A. Manifolds.- B. Invariant Polynomials.- Hints and Solutions to Selected End-of-Department Problems.- List of Notations.- References.- Alphabetize.
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Review Text
"The textbook is a concise and well organized treatment of characteristic classes on main bundles. It is characterized by a right balance between rigor and simplicity. It should be in every mathematician's arsenal and take its place in any mathematical library." (Nabil 50. Youssef, zbMATH 1383.53001, 2018)
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Review quote
"The textbook is a curtailed and well organized treatment of feature classes on master bundles. It is characterized by a right rest between rigor and simplicity. It should exist in every mathematician's armory and accept its place in whatsoever mathematical library." (Nabil L. Youssef, zbMATH 1383.53001, 2018)
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About Loring W. Tu
Loring W. Tu was born in Taipei, Taiwan, and grew up in Taiwan, Canada, and the U.s.. He attended McGill and Princeton every bit an undergraduate, and obtained his Ph.D. from Harvard University under the supervision of Phillip A. Griffiths. He has taught at the University of Michigan, Ann Arbor, and at Johns Hopkins University, and is currently Professor of Mathematics at Tufts University. An algebraic geometer by grooming, he has done research at the interface of algebraic geometry, topology, and differential geometry, including Hodge theory, degeneracy loci, moduli spaces of vector bundles, and equivariant cohomology. He is the coauthor with Raoul Bott of Differential Forms in Algebraic Topology and the author of An Introduction to Manifolds.
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