An Introduction to Manifolds

by Loring W. Tu

Manifolds, the higher-dimensional analogs of smooth curves and surfaces, are fundamental objects in modern mathematics. Combining aspects of algebra, topology, and analysis, manifolds have also been applied to classical mechanics, general relativity, and quantum field theory. In this streamlined introduction to the subject, the theory of manifolds is presented with the aim of helping the reader achieve a rapid mastery of the essential topics. By the end of the book the reader should be able to compute, at least for simple spaces, one of the most basic topological invariants of a manifold, its de Rham cohomology. Along the way, the reader acquires the knowledge and skills necessary for further study of geometry and topology. The requisite point-set topology is included in an appendix of twenty pages; other appendices review facts from real analysis and linear algebra. Hints and solutions are provided to many of the exercises and problems. This work may be used as the text for a one-semester graduate or advanced undergraduate course, as well as by students engaged in self-study. Requiring only minimal undergraduate prerequisites, 'Introduction to Manifolds' is also an excellent foundation for Springer's GTM 82, 'Differential Forms in Algebraic Topology'.

Genres: Mathematics, Textbooks, Geometry, Nonfiction, Algebra, Science

428 pages, Paperback

First published October 29, 2007

Reviews

[Erickson · Rating 5 out of 5]

Used this extensively for my differential geometry needs (general relativity, gauge theory, etc.) and especially useful for my mathematical gauge theory course I took earlier this year. I did not reach Chapter 7 (de Rham theory) though I read bits of it, and my background in algebraic topology (a course-worth of it) tells me that this chapter is useful as a prototypical example of cohomology theory ---de Rham--- where examples are very easy to grasp in comparison to other (co)homology theories.

If I have to choose, I would say this book is to me better than Lee. It is detailed, easy to understand, and even containing some surprises (like notion of graded algebra, derivations, which are typically glossed over in many courses on differential geometry for good reasons). Some might find earlier chapter a waste of time ("toy differential geometry in Euclidean space") but I think this is very useful especially for people who are not pure mathematicians. A physics student could easily learn pure mathematics from this book thanks to the earlier chapters.

Now, if you skip straight to Section 5 where manifold theory begins proper, this is still a very good book. In fact, I feel that Lee's Smooth Manifolds is a "reference text" because for many purposes (especially introduction) it's overkill. It is very helpful when you want to look for technical details that you might not appreciate when you first learn this (e.g. fibre bundle construction lemma). So as a "textbook", this is the better book for me. Of course, for me I also treat this as a reference text because it actually has enough details for most purposes I need, and only in some rare occasion e.g. proving the manifold construction lemma that I need Lee instead of Tu's text.

In short, if you want to get basic differential geometry right and you only have enough funds to get a book, get Tu.

[Grayson Croom · Rating 4 out of 5]

Loring Tu develops the theory of smooth manifolds gently and rapidly in this text. Often this comes at the expense of showing interesting uses of the machinery built, which can feel unsatisfying. However, I found this book to be very helpful in getting my foot in the door.

Other disjointed thoughts:

I am glad the book included a chapter on the de Rham cohomology and now look forward to learning about homological algebra (diagram chasing is fun!)

I would have appreciated more exposition on vector bundles, but this is largely a non-issue as one can supplement with LeeSM.

Great book.

[Ezra · Rating 5 out of 5]

Lucid writing. Exercises tend on the easy side. I would warn the prospective reader that the book is very introductory compared to books often compared, such as Lee, and does not cover the same depth or breadth of material.

[Mark Moon · Rating 5 out of 5]

A very nice book. Covers plenty of ground (for an introduction) without every trying to go too fast. I hope I get to teach out of it someday.