國立高雄大學圖資館 |

Language: English

Back

Differential geometryconnections, cu...

SpringerLink (Online service)

Differential geometryconnections, curvature, and characteristic classes /

Record Type:	Electronic resources : Monograph/item
Title/Author:	Differential geometryby Loring W. Tu.
Reminder of title:	connections, curvature, and characteristic classes /
Author:	Tu, Loring W.
Published:	Cham :Springer International Publishing :2017.
Description:	xvii, 347 p. :ill. (some col.), digital ;24 cm.
Contained By:	Springer eBooks
Subject:	Geometry, Differential.
Online resource:	http://dx.doi.org/10.1007/978-3-319-55084-8
ISBN:	9783319550848$q(electronic bk.)

Differential geometryconnections, curvature, and characteristic classes /
Tu, Loring W.

Differential geometryconnections, curvature, and characteristic classes /[electronic resource] :by Loring W. Tu. - Cham :Springer International Publishing :2017. - xvii, 347 p. :ill. (some col.), digital ;24 cm. - Graduate texts in mathematics,2750072-5285 ;. - Graduate texts in mathematics ;129..

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 Gauss-Bonnet 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. Vector-Valued 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 End-of-Section Problems -- List of Notations -- References -- Index.

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 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 self-contained, 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 far-fetched to argue that differential geometry should be in every mathematician's arsenal.

ISBN: 9783319550848$q(electronic bk.)

Standard No.: 10.1007/978-3-319-55084-8doiSubjects--Topical Terms:

182610
Geometry, Differential.

LC Class. No.: QA641

Dewey Class. No.: 516.36

Differential geometryconnections, curvature, and characteristic classes /
LDR:04796nmm a2200325 a 4500 001 517852
003 DE-He213
005 20180117150608.0
006 m d
007 cr nn 008maaau
008 180316s2017 gw s 0 eng d
020 $a 9783319550848$q(electronic bk.)
020 $a 9783319550824$q(paper)
024 7 $a 10.1007/978-3-319-55084-8 $2 doi
035 $a 978-3-319-55084-8
040 $a GP $c GP
041 0 $a eng
050 4 $a QA641
072 7 $a PBMP $2 bicssc
072 7 $a MAT012030 $2 bisacsh
082 0 4 $a 516.36 $2 23
090 $a QA641 $b .T883 2017
100 1 $a Tu, Loring W. $3 275682
245 1 0 $a Differential geometry $h [electronic resource] : $b connections, curvature, and characteristic classes / $c by Loring W. Tu.
260 $a Cham : $b Springer International Publishing : $b Imprint: Springer, $c 2017.
300 $a xvii, 347 p. : $b ill. (some col.), digital ; $c 24 cm.
490 1 $a Graduate texts in mathematics, $x 0072-5285 ; $v 275
505 0 $a 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 Gauss-Bonnet 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. Vector-Valued 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 End-of-Section Problems -- List of Notations -- References -- Index.
520 $a 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 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 self-contained, 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 far-fetched to argue that differential geometry should be in every mathematician's arsenal.
650 0 $a Geometry, Differential. $3 182610
650 1 4 $a Mathematics. $3 184409
650 2 4 $a Differential Geometry. $3 273785
650 2 4 $a Algebraic Geometry. $3 274807
710 2 $a SpringerLink (Online service) $3 273601
773 0 $t Springer eBooks
830 0 $a Graduate texts in mathematics ; $v 129. $3 436082
856 4 0 $u http://dx.doi.org/10.1007/978-3-319-55084-8
950 $a Mathematics and Statistics (Springer-11649)