Differential Geometry : Connections, Curvature, and Characteristic Classes / by Loring W. Tu.

Format:

Book

Author/Creator:

Tu, Loring W., Author.

Series:

Graduate Texts in Mathematics, 2197-5612 ; 275

Language:

English

Subjects (All):

Geometry, Differential.

Geometry, Algebraic.

Differential Geometry.

Algebraic Geometry.

Local Subjects:

Differential Geometry.

Algebraic Geometry.

Physical Description:

1 online resource (xvii, 347 pages) : illustrations (some color)

Edition:

1st ed. 2017.

Place of Publication:

Cham : Springer International Publishing : Imprint: Springer, 2017.

Language Note:

English

Summary:

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.

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 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.

Notes:

Includes bibliographical references and index.

ISBN:

3-319-55084-5

9783319550848 (ebook)

OCLC:

989727720