Differential geometry is a necessary tool for anyone interested in advanced physics, but it can be an intimidating subject to learn. Using it requires understanding that goes beyond what can be easily described in pictures, and the mathematics often gets opaque quickly. The subject usually involves large formulas and complex mathematics with its own conventions (such as the index manipulation in tensors).

This book covers the essentials of differential geometry up to relativity (and more). It takes the reader from a basic introduction to advanced topics.

The book is less than 230 pages long, so the information presentation is terse, almost telegraphic. Chapters include:

(1) “Introduction”;

(2) “Manifolds,” which covers the essential definitions of a manifold and functions on a manifold;

(3) “Vector Fields and One-Form Fields”;

(4) “Basis Fields,” including changes of basis and commutators;

(5) “Integration,” including higher-dimensional integration and a discussion (but not a proof) of Stokes’ theorem;

(6) “Over a Map,” which covers basis fields over a map and one-form fields;

(7) “Directional Derivatives,” on the Lie derivative and parallel transport;

(8) “Curvature,” on torsion and other transports;

(9) “Metrics,” including basic metrics and general relativity;

(10) “Hodge Star and Electrodynamics,” covering the wave equation and electrodynamics; and

(11) “Special Relativity,” including Lorentz transformations and the twin paradox.

There are also three appendices: one on the Scheme language (enough to make the code readable), one on the notation used by the authors (with some of the specific Scheme conventions used in the book), and one on tensors and how they are represented in the code. There are a number of exercises scattered throughout the book.

What makes this book unique is that, in addition to the standard mathematical notation, the concepts are also given with Scheme code. At any point, interested readers can run the Scheme functions using an MIT-Scheme interpreter. Most of the code (though not all) is available for download, although the uniform resource locator (URL) only appears in the references section and code is less well represented in the later chapters.

The reader will ideally know a bit about differential geometry when starting, and have at least some familiarity with manifolds, vector fields, differential forms, and wedge products. Readers with no more than this basic knowledge will find that the text requires careful reading and very close study. It will probably also be helpful to know a bit of Scheme, although the functions given are rarely complicated in the programming sense, beyond nesting definitions and temporary assignments. There are few loops and even fewer if statements. The examples do use many higher-order functions, so familiarity with this type of programming will be more than helpful.

Most of the Scheme code samples are short. Few are longer than 20 lines, but they pack a lot of information and merit close study. It will probably also help the reader to have read and worked through Sussman and Wisdom’s previous work [1], which provides a bit more information on how the code works, as well as sharing a code base.

Interested readers with some background in a classical approach to differential geometry are likely to find this book interesting and potentially quite enlightening. However, even those with a strong background in differential geometry should expect to take some time, since the Scheme code is likely to take some study. I would not recommend it for readers just starting out with differential geometry. The text is likely to be intimidating, although it is certainly possible to work through the book by running each code example. Having a more expository traditional text (or two) for reference is recommended.