Introduction to Geometry and Topology (Compact Textbooks in Mathematics) by Werner Ballmann (PDF)

Description

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.

User’s Reviews

Editorial Reviews: Review “This book is an excellent companion to everyone involved in courses on Differential Topology and/or Differential Geometry. Quite efficiently it gets to several deep results avoiding diversions, still giving a sense of pace friendly to the reader. Otherwise it takes matters till the accesible point and proposes literature for further progress. Strongly recommendable.” (Jesus M. Ruiz, European Mathematical Society, euro-math-soc.eu, February, 2019)“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. Many will 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) From the Back Cover 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. About the Author Werner Ballmann is Professor of Differential Geometry at the University of Bonn and Director at the Max Planck Institute for Mathematics in Bonn. Read more

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐(4.5 stars out of 5)This book contains a solid introduction to the subject of Differential Topology and Differential Geometry, and even starts out with a digestible chapter on standard topology – something that I hardly ever see in larger-sized textbooks apart from a few ideas relegated to an appendix, more or less a “compact series” textbook like this one (and it *is* compact at just over 150 pages!).The remaining chapters focus on manifolds, fibers and homology theory, and provide enough material for a geometry-focused early graduate course – one semester if topology has been covered, and two semesters (with some supplementation from the instructor) if not. The weakest point is likely the small question selection; I believe it is Chapter Three that hardly has any. Additionally, the first section on Topology is useful for developing a geometric point of view, and should not be used by anyone learning point-set topology for the first time. It should also be noted that this book is likely not suited for the first-timer in graduate or upper undergraduate mathematics, as the compact size will give more of a sense of reading someone’s notes from a previous course, rather than a flexed-out textbook. That being said, the amount that is here is sufficient for anyone experienced in proof-based courses and with working knowledge of “baby reals” and point-set topology to digest, and it avoids the long-winded descriptions, classical development (provided one is not looking for this), or over-reliance on algebra or algebraic geometry.All in all, however, a good textbook and a great resource for a professional to have while away from the office or the home study area. It should not be one’s only reference, however.

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