Author:  Tristan Needham
Publisher: Princeton
Pages: 584
ISBN: 978-0691203706
Print: 0691203709
Kindle: B08TT6QBZH
Audience: Math enthusiasts
Rating: 5
Reviewer: Mike James
The best math book I have read in a long time...

The ideal reader for this book is anyone who has been subjected to a course in differential geometry or forms. Traditional courses and books tend to give you the algebra without worrying about what it all means. This book is focused on getting you to think about the geometry and its relationship to the algebra.

Why should you want to learn about differential geometry?

My best answer is that it is interesting and has many amazing applications that makes thinking about the world much easier. Differential geometry is about the idea of curved spaces or manifolds. We live on the curved surface of the earth and yet many of us have no idea what the consequences of this are - hurricanes wouldn't form if we lived on a flat earth and our weather patterns would be dramatically different.

Ask almost anyone what happens if you start off at a point on the surface of the earth pointing in a given direction, travel around a circuit back to their original position keeping pointing in the same direction. The question is when they get back are they still pointing in the same direction? Of course, they are - only they aren't. Moving around a loop in a curved space causes a rotation - holonomy. Flat earthers don't need a trip up into space to discover that we live on curved surface - just a trip around a ground-based loop.

Differential geometry is also the basis of special and general relativity and may have a huge role to play in the way that we understand neural networks and many more situations where thinking in curves is a good approach.

The subtitle of this book is "A Mathematical Drama in Five Acts" and this gives you some idea of the style. The presentation is about as far from "theorem-proof" as you can get. The "Visual" in the title also indicates that there are are lots of diagrams and thinking geometrically is a skill you will either have already or develop while reading.

The first act is The Nature of Space and is a very good introduction to non-Euclidean geometry. It introduces the basic idea of curvature and the Gauss-Bonnet theorem which is a major player in the whole show.

Act II is The Metric and, yes, you get to meet the basic idea of the idea of distance in a curved space, but not in the abstract linear algebra or tensor-based way it is usually introduced. We also meet some complex analysis in the form of conformal maps and Mobius transforms and at last I understand not just that they are important, but what makes them important.

Act III is central to the performance - Curvature. Starting from plane curves, which are easy to think about, the book extends the idea to 3D. We move on to meet the principle ideas of differential geometry - geodesics, the spherical map and the Gauss-Bonnet theorem in all its forms and proofs. The repeated proofs really do get you to see the different meanings and interpretations of what the theorem is all about. The section on Bagels really helps.

Act IV makes the idea of parallel transport seem essential and yet footnotes reveal the fact that so much of Einstein's development of general relativity came before the idea was thought of - how is this possible? This is the act in which we learn about holonomy and meet yet more proofs of the Gauss-Bonnet theorem. The final parts introduce the idea that curvature is force and vice versa. After looking at Riemann curvature we derive the Einstein equation in a way that is more directly physical than any I have encountered. It's all a matter of thinking about the tidal forces on particles and it makes it all seem very reasonable rather than just "pulled out of a hat and shown to work". If you are wanting to understand general relativity you have to read this to gain an insight into why it follows from physical observation rather than just mathematical beauty.

The final act is about forms. Why should you want to know about forms? A long time ago I was told "forms are what we integrate" and I was left wondering why for too many years. Now I'm happy that forms are natural and my preference for all things symmetric has been replaced by an appreciation of the skew symmetric.  This act explains the world of forms, dual spaces, tensors and why forms are confused with vectors. I now also understand why the Faraday and Maxwell tensors are natural representations rather than just appreciating that they work. And finally the integration part of forms and the classical integral theories are made to seem obvious.

I took about four months to read this book and I was sad when I finished it. So sad I started reading parts of it again until my copy of Visual Complex Analysis arrives - the author's first book in the same style.  The book is never less than engaging with lots of historical asides and footnotes. If I have to find a complaint then I'd prefer the footnotes were boxouts so I could read them more as part of the flow. I also wish there was a glossary of symbols because occasionally when I picked it up after even a short break I sometimes couldn't remember what a symbol was supposed to represent.

The diagrams are very helpful, but only if you use them as a nudge to get your own mental imagery going - flat 2D representations of processes in 3D space are never going to be enough unless you can internalize the vision. What is most important about this book is that it isn't a presentation of differential geometry and forms, it is an explanation. We need more math books like this.

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