Overview
- For undergraduates! Required background material is typically covered in the first 2 or 3 years of university
- The role of point set topology is kept to a minimum
- Theory of manifolds appears as a natural continuation of multivariable calculus
Part of the book series: Springer Undergraduate Mathematics Series (SUMS)
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Table of contents (9 chapters)
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Front Matter
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Back Matter
Keywords
- introduction to manifolds textbook
- smooth functions on manifolds
- tangent spaces
- differential forms textbook
- integration of differential forms
- undergraduate differential geometry
- textbook differential geometry
- properties manifolds
- vector bundles
- integration
- tangent spaces
- submanifolds
- manifolds
- vector fields
About this book
This textbook serves as an introduction to modern differential geometry at a level accessible to advanced undergraduate and master's students. It places special emphasis on motivation and understanding, while developing a solid intuition for the more abstract concepts. In contrast to graduate level references, the text relies on a minimal set of prerequisites: a solid grounding in linear algebra and multivariable calculus, and ideally a course on ordinary differential equations. Manifolds are introduced intrinsically in terms of coordinate patches glued by transition functions. The theory is presented as a natural continuation of multivariable calculus; the role of point-set topology is kept to a minimum.
Questions sprinkled throughout the text engage students in active learning, and encourage classroom participation. Answers to these questions are provided at the end of the book, thus making it ideal for independent study. Material is further reinforced with homework problems ranging from straightforward to challenging. The book contains more material than can be covered in a single semester, and detailed suggestions for instructors are provided in the Preface.
Reviews
Authors and Affiliations
About the authors
Gal Gross is a Ph.D. student in mathematics at the University of Toronto, working in combinatorics and algebra with a special interest in additive combinatorics. Gross' other mathematical interests include differential geometry, set theory and foundational questions.
Eckhard Meinrenken is a professor of mathematics at the University of Toronto, working in the fields of differential geometry and mathematical physics. His contributions include a proof of the Guillemin-Sternberg conjecture in symplectic geometry and the development, with Alekseev and Malkin, of the theory of group-valued momentum maps. In 2002 he was an invited speaker at the ICM in Beijing, and in 2008 he was elected Fellow of the Royal Society of Canada. Meinrenken's book Clifford Algebras and Lie Theory was published (c) 2013 in Springer's Ergebnisse series
Bibliographic Information
Book Title: Manifolds, Vector Fields, and Differential Forms
Book Subtitle: An Introduction to Differential Geometry
Authors: Gal Gross, Eckhard Meinrenken
Series Title: Springer Undergraduate Mathematics Series
DOI: https://doi.org/10.1007/978-3-031-25409-3
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
Softcover ISBN: 978-3-031-25408-6Published: 26 April 2023
eBook ISBN: 978-3-031-25409-3Published: 25 April 2023
Series ISSN: 1615-2085
Series E-ISSN: 2197-4144
Edition Number: 1
Number of Pages: XIV, 343
Number of Illustrations: 73 b/w illustrations, 7 illustrations in colour
Topics: Topology, Differential Geometry