# Differential Geometry

## Connections, Curvature, and Characteristic Classes

Textbook

Part of the Graduate Texts in Mathematics book series (GTM, volume 275)

1. Front Matter
Pages i-xvii
2. Loring W. Tu
Pages 1-69
3. Loring W. Tu
Pages 71-94
4. Loring W. Tu
Pages 95-150
5. Loring W. Tu
Pages 151-198
6. Loring W. Tu
Pages 199-239
7. Loring W. Tu
Pages 241-291
8. Back Matter
Pages 293-347

### Introduction

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.

### Keywords

Christoffel symbols Codazzi–Mainardi equation Gauss Curvature equation Gauss Theorema egregium Gauss–Bonnet theorem Gram–Schmidt process

#### Authors and affiliations

1. 1.Department of MathematicsTufts UniversityMedfordUSA

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.

### Bibliographic information

• Book Title Differential Geometry
• Book Subtitle Connections, Curvature, and Characteristic Classes
• Authors Loring W. Tu
• Series Title Graduate Texts in Mathematics
• Series Abbreviated Title Graduate Texts Mathematics
• DOI https://doi.org/10.1007/978-3-319-55084-8
• Copyright Information Springer International Publishing AG 2017
• Publisher Name Springer, Cham
• eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
• Hardcover ISBN 978-3-319-55082-4
• Softcover ISBN 978-3-319-85562-2
• eBook ISBN 978-3-319-55084-8
• Series ISSN 0072-5285
• Series E-ISSN 2197-5612
• Edition Number 1
• Number of Pages XVII, 347
• Number of Illustrations 72 b/w illustrations, 15 illustrations in colour
• Topics
• Buy this book on publisher's site

## Reviews

“The textbook is a concise and well organized treatment of characteristic classes on principal 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 L. Youssef, zbMATH 1383.53001, 2018)