Manifolds Vector Fields and Differential Forms: An Introduction to Differential Geometry

Name: Manifolds Vector Fields and Differential Forms: An Introduction to Differential Geometry
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ISBN13: 9783031254086
Binding: Paperback
Subject: Mathematics and Statistics
Publisher: Springer Verlag
Publisher Imprint: Springer
Publication Date: December 10, 2023
Pages: 343
Original Price: EUR 49.99
Language: English
Edition: N/A
Item Weight: 610 grams
BISAC Subject(s): Topology

About The 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.

About The Author

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

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rom Tangent Spaces to Stokes’ Theorem: Exploring Gross’s Introduction to Differential Geometry

Gal Gross’s book is a thoughtfully organized entry point into differential geometry, written for readers with background in multivariable calculus and linear algebra. It develops the subject step by step, each chapter adding a new layer of abstraction while maintaining geometric intuition.

Chapter-wise Breakdown

Manifolds – Introduces smooth manifolds, charts, atlases, and examples (spheres, projective spaces).

Tangent Spaces and Vector Fields – Defines tangent vectors, derivations, and vector fields, including flows and Lie brackets.

Differential Forms – Builds exterior algebra, wedge products, exterior derivative, and pullbacks.

Integration – Covers orientation, integration of differential forms, and the statement/proof of Stokes’ theorem.

Submanifolds and Immersions – Examines embeddings, immersions, and submanifolds with examples.

Vector Bundles – Introduces bundles, sections, and basic constructions, preparing for more advanced geometry.

Comment

The text balances formal proofs with clear explanations and motivating examples. Its strength lies in accessibility: abstract machinery is always tied to concrete geometric objects. The progression from manifolds to Stokes’ theorem feels natural, making the book a strong choice for advanced undergraduates, beginning graduate students, or self-learners seeking a solid conceptual foundation in differential geometry.

Manifolds Vector Fields and Differential Forms: An Introduction to Differential Geometry

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