Manifolds, Vector Fields, and Differential Forms

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.

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

Auteur

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

Résumé
"This book is intended to be a modern introduction to the basics of differential geometry, accessible to undergraduate and master students. From my point of view, this goal is achieved, the book being very well structured and supported by illustrative examples and problems. ... this book will be of great interest for undergraduate students, master students, and also helpful for instructors." (Gabriel Eduard Vilc, zbMATH 1522.53001, 2023)

Contenu

1. Introduction.- 2. Manifolds.- 3. Smooth maps.- 4. Submanifolds.- 5. Tangent spaces.- 6. Vector fields.- 7. Differential forms.- 8. Integration.- 9. Vector bundles.- Notions from set theory.- Notions from algebra.- Topological properties of manifolds.- Hints and answers to in-text questions.- References.- List of Symbols.- Index.