Differential Forms and Connections

Rating:
Summary: Not for everyone, flawed in basic ways
Review: I disagree with reviewers who found this book useful for self-study. I would not recommend it for individuals first learning this material. The book is frankly contradictory in places, and frustratingly repetitive in others. In the early chapters it assumes concepts not yet explained, and introduces terminology and symbols that are nowhere defined.

If you already know quite a bit, you may find this approach enlightening. But if you're just beginning to master these concepts, I suggest you look elsewhere.

I also suggest that much tighter editing would do this book a world of good. Go with Kreiszig, or Lovelock and Rund instead.

Rating:
Summary: The ONLY book really suited for self study
Review: I would just like to point out that Darling's book is the only book I've encountered which is suited for self study. It resembles someone's classroom notes - i.e., nothing fancy, no glossy color 3-d graphics or such - but it is very neatly organized, with many examples and helpful problems, and it is really, really suited for someone trying to study the subject by him/herself (me ... ). It is not very physically oriented - not many physical examples are provided throughout the text, and it is mathematical in nature, but don't let that deter you! In fact, the sharp distinction between mathematics and physics is pedagogically wise.
Another good thing about this book is that it does not begin with completely abstract definitions. First of all it develops exterior calculus and diff. manifolds in ordinary Euclidian space. This is a must for anyone studying on their own, believe me! No matter how mathematically mature you are, those things just don't make sense unless you've seen how they work in familiar settings. You don't have to worry, though - Darling keeps his notation clean; Darling tries as hard as he can to keep everything in pure geometrical language, referring to a specific basis only when absolutely necessary (or when it helps one understand).
I cannot say how good a classroom text this is, but do yourself a favor and check it out if you're thinking of studying this on your own! Darling is a clear and (equally important!) responsible teacher.

Rating:
Summary: A must for both the physicist and mathematician
Review: RWR Darling should be the first and foremost book for learning about differential geometry both for physicists and mathematicians. I have learned from numerous books on this subject, and while I can't say Darling includes everything one could want (I can't say anyone ever does), his text explains some very esoteric ideas in terms of linear algebra and vector calculus.

A notable departure this book makes is dispensing with the usual coordinate basis for tangent spaces which is commonly used by physicists. To the experienced physics reader, this may seem daunting, and unnecessarily abstract at first. However, the pay-off in the ability later on to discuss gauge theories and fiber bundles is huge.

This book is also suited for mathematicians less interested in physics. Darling does not always assume that a manifold has some metric, and discusses the subtle differences between vectors and co-vectors in modern mathematical language. Secondly, he provides a lot of motivation for the mathematical constructions and takes great care to present key definitions in extremely coordinate free ways.

Rating:
Summary: Gauge theories in the mathematical way
Review: The main difficulty found by physicists in the learning of modern differential geometry is topology. The various constructions introduced by Cartan and others, differential forms, connections, even fiber bundles, on the contrary, pose no difficulties: it is only a question of developing the appropriate muscles and reflexes. R. Darling wrote the ideal book to teach connections on a G-bundle (gauge theories, in the nomenclature of physicists), by refraining, as much as possible, to use explicit topology. As physicists are not a special kind of human beings, I believe what I said above is also true of (beginning) mathematicians. Otherwise, why would Darling choose such course (in the navigational sense). The book starts with Cartan calculus in Euclidean space, continues there up to surface theory, then introduces (intrinsic) manifolds. Perhaps the key concept of the book comes next: Vector Bundles. All previous constructions are extended to bundles, and the concept of conn! ections on vector bundles deserves a special chapter. The book ends with Applications to Gauge Field Theory (mathematics-wise, but quite accessible). There are many pedagogical virtues in this much welcome book. Finally a good alternative to Bishop-Goldberg`s "Tensor Calculus on Manifolds".

Rating:
Summary: Excellent book
Review: This is a very modern, very concise, and very efficient book. By using vector bundles the curvature forms on semi-Riemannian manifolds are introduced. Definitions are given clearly and intuitively. Without spending tons of pages on digression to minimal surfaces, Hopf-Rinow thm, Gauss-Bonnet thm, etc., the book builds enough machinery to describe the gauge field theory in the last chapter. Most other differential geometry books either throw in too many applications to waste reader's time or give vague definitions (too bluntly abstract or not self-contained) to confuse the reader.

All exercise problems are interesting and important. Hints are given to some of them.

I found Warner's "Foundations of Differentiable Manifolds and Lie Groups" is a good complement to address the algebraic and topological side of differential geometry.

Rating:
Summary: Worthy
Review: This text covers many topics and is quite approachable for a beginner but requires very careful reading. The author tends to express important definitions in the flow of the text inconveniencing references to said definitions from later sections. This book does not require and makes little appeal to notions from topology which renders the text more accessible but short changes the reader of important insights.

The definition of tangents and tangent space are introduced rather awkwardly; the set of continuously differentiable functions (C-infinity) is dumped on the readers lap without preliminary discussion. These issues are addressed by "Tensor analysis on manifolds" by Bishop and Goldberg though the latter book is _very_ terse.

I recommend the subject text but suggest that one simultaneously read "Tensor analysis on manifolds" by Bishop and Goldberg. I also recommend that the beginner/self studier consider reading either "Differential Geometry" by Kreyszig or possibly "Differential Forms With Applications to the Physical Sciences" by Harley Flanders before reading either of the said texts. Kreyszig does a much better job of actually writing a readable text (the notation used is a bit old though). Ideally the Bishop and Goldberg texts should be combined and rewritten in the Kreyszig style.