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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 About this Textbook

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.
 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.
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Bibliographic Information
 Bibliographic Information

 Book Title
 Differential Geometry
 Book Subtitle
 Connections, Curvature, and Characteristic Classes
 Authors

 Loring W. Tu
 Series Title
 Graduate Texts in Mathematics
 Series Volume
 275
 Copyright
 2017
 Publisher
 Springer International Publishing
 Copyright Holder
 Springer International Publishing AG
 eBook ISBN
 9783319550848
 DOI
 10.1007/9783319550848
 Hardcover ISBN
 9783319550824
 Series ISSN
 00725285
 Edition Number
 1
 Number of Pages
 XVII, 347
 Number of Illustrations and Tables
 72 b/w illustrations, 15 illustrations in colour
 Topics