Overview
- Lecture-tested introduction to topology, differential topology, and differential geometry
- Contributes to a wide range of topics on a few pages
- About 70 exercises motivate the application of the learned field
- Contains valuable hints for further reading
Part of the book series: Compact Textbooks in Mathematics (CTM)
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Table of contents (4 chapters)
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Front Matter
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Back Matter
Keywords
About this book
This book provides an introduction to topology, differential topology, and differential geometry. It is based on manuscripts refined through use in a variety of lecture courses. The first chapter covers elementary results and concepts from point-set topology. An exception is the Jordan Curve Theorem, which is proved for polygonal paths and is intended to give students a first glimpse into the nature of deeper topological problems.
The second chapter of the book introduces manifolds and Lie groups, and examines a wide assortment of examples. Further discussion explores tangent bundles, vector bundles, differentials, vector fields, and Lie brackets of vector fields. This discussion is deepened and expanded in the third chapter, which introduces the de Rham cohomology and the oriented integral and gives proofs of the Brouwer Fixed-Point Theorem, the Jordan-Brouwer Separation Theorem, and Stokes's integral formula.
The fourth and final chapter is devoted to the fundamentals of differential geometry and traces the development of ideas from curves to submanifolds of Euclidean spaces. Along the way, the book discusses connections and curvature--the central concepts of differential geometry. The discussion culminates with the Gauß equations and the version of Gauß's theorema egregium for submanifolds of arbitrary dimension and codimension.
This book is primarily aimed at advanced undergraduates in mathematics and physics and is intended as the template for a one- or two-semester bachelor's course.
Reviews
“Mathematical exposition has a curatorial aspect. … Along the path to the Stokes theorem, readers meet some undisguised algebraic topology and beyond it get a solid introduction to differential geometry, including the Riemann curvature tensor. Manywill wish they had used this book as students. Summing Up: Recommended. Lower-division undergraduates through faculty and professionals.” (D. V. Feldman, Choice, Vol. 56 (6), February, 2019)
Authors and Affiliations
About the author
Bibliographic Information
Book Title: Introduction to Geometry and Topology
Authors: Werner Ballmann
Translated by: Walker Stern
Series Title: Compact Textbooks in Mathematics
DOI: https://doi.org/10.1007/978-3-0348-0983-2
Publisher: Birkhäuser Basel
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Basel 2018
Softcover ISBN: 978-3-0348-0982-5Published: 28 July 2018
eBook ISBN: 978-3-0348-0983-2Published: 18 July 2018
Series ISSN: 2296-4568
Series E-ISSN: 2296-455X
Edition Number: 1
Number of Pages: X, 169
Number of Illustrations: 8 b/w illustrations, 20 illustrations in colour
Topics: Manifolds and Cell Complexes (incl. Diff.Topology), Differential Geometry, Global Analysis and Analysis on Manifolds