Graduate Texts in Mathematics

Differential Geometry

Connections, Curvature, and Characteristic Classes

Authors: Tu, Loring W.

  • Narrative provides a panorma of some of the high points in the history of differential geometry
  • Problems are presented in each chapter with selected solutions and hints given at the end of the book
  • Accessible to graduate students of mathematics and physics
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  • ISBN 978-3-319-55084-8
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About this Textbook

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.

About the authors

Loring W. Tu was born in Taipei, Taiwan, and grew up in Taiwan, Canada, and the United States. He attended McGill and Princeton as 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 training, 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.

Table of contents (6 chapters)

  • Chapter 1 Curvature and Vector Fields

    Tu, Loring W.

    Pages 1-69

  • Chapter 2 Curvature and Differential Forms

    Tu, Loring W.

    Pages 71-94

  • Chapter 3 Geodesics

    Tu, Loring W.

    Pages 95-150

  • Chapter 4 Tools from Algebra and Topology

    Tu, Loring W.

    Pages 151-198

  • Chapter 5 Vector Bundles and Characteristic Classes

    Tu, Loring W.

    Pages 199-239

Buy this book

eBook n/a
  • ISBN 978-3-319-55084-8
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
Hardcover n/a
  • ISBN 978-3-319-55082-4
  • Free shipping for individuals worldwide
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Bibliographic Information

Bibliographic Information
Book Title
Differential Geometry
Book Subtitle
Connections, Curvature, and Characteristic Classes
Authors
Series Title
Graduate Texts in Mathematics
Series Volume
275
Copyright
2017
Publisher
Springer International Publishing
Copyright Holder
Springer International Publishing AG
eBook ISBN
978-3-319-55084-8
DOI
10.1007/978-3-319-55084-8
Hardcover ISBN
978-3-319-55082-4
Series ISSN
0072-5285
Edition Number
1
Number of Pages
XVII, 347
Number of Illustrations and Tables
72 b/w illustrations, 15 illustrations in colour
Topics