Differential Forms and Connections - R. W. R Darling

This is the first book on differential geometry that I've read. It's rather different to other maths books I've read - I've got the impression that it's typeset in Word rather than LaTeX, for example! More concretely, it's certainly towards the applied end of the spectrum. It doesn't feel terribly formalised, and there are plenty of concrete examples in 3D space.

I read the book as a maths tourist - didn't do the exercises, and skipped anything I didn't get without worrying. The vast majority of the content can be understood in terms of similarity to a stitched-together R^n set-up, arranged so that it's coordinate-insensitive. Differential forms can be mostly understood in terms of the usual symbol-plugging extended to exterior algebra, and an awful lot can run off intuition.

So, what's the subject? Basically, it's doing calculus on spaces made of stitched together R^ns in a coordinate-insensitive way (yes, I know I already said that). The first chapter introduces exterior algebra. This is kind of the the extension of vectors to multi-dimensional vectors, and is a great way of putting div, curl and grad into a framework that makes sense in higher dimensions. After that, it introduces a whole slew of bits and pieces - differential manifolds, vector bundles, metrics, etc., ending up with integration in Chapter 8. Throughout this, the combination of examples and intuition work well.

Then there was, for me, some weird gear change. Chapter 9 introduced connections, and I lost the plot entirely. Chapter 10 would have brought in a whole pile of very interesting-looking physics, but it was all heavily dependent on connections, so I'd basically lost the plot.

Where did I lose it? I think this was implicit in the way surface understanding and proper understanding of the subject fit together. While gluing the symbols together mostly behaves pretty intuitively, it's easy to miss the formalisation that's being attempted. A lot of the symbols are representing functions on functions, and if you're running at the surface level things suddenly get rather difficult when you ask 'what's really going on here?'.

Since most components of the subject seem to be made out of a) vector spaces and linear transformations b) diffeomorphisms and c) function compositions, I have this suspicion that once the effort has been put in and the underlying formalism explored, the whole thing would become relatively simple again. As so much of the subject is function composition, I'm very tempted to try to reimplement various chunks of the book as code, and see if functional programming can help me understand the abstractions. Sadly, my coding time is more than a little limited at the moment. One day, perhaps.

Back to this book: Is it any good? Dunno - it's the only book on differential geometry I've read! I'm pretty sure that a book by e.g. Lang would make me much more confident about how it glues together formally, but would rather destoy any intuition I had. This book does focus on the intuition, but perhaps at too much expense to the other aspects. Perhaps a combination of this book plus another would be optimal?

Posted 2011-04-24.