Overview
- 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
Part of the book series: Graduate Texts in Mathematics (GTM, volume 275)
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Table of contents (6 chapters)
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About this book
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.
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Bibliographic Information
Book Title: Differential Geometry
Book Subtitle: Connections, Curvature, and Characteristic Classes
Authors: Loring W. Tu
Series Title: Graduate Texts in Mathematics
DOI: https://doi.org/10.1007/978-3-319-55084-8
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer International Publishing AG 2017
Hardcover ISBN: 978-3-319-55082-4Published: 15 June 2017
Softcover ISBN: 978-3-319-85562-2Published: 01 August 2018
eBook ISBN: 978-3-319-55084-8Published: 01 June 2017
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: Differential Geometry, Algebraic Geometry