I’m aware of only two books whose central theme is the application of Cartan moving frames to other areas of mathematics. Both books fall within the AMS Graduate Studies series; they have similar titles and their front covers are almost identical. The first book, published in 2003, was* Cartan for Beginners; Differential Geometry via Moving Frames and Exterior Differential Systems* by Ivey and Landsberg. The second book is, of course, this one by Jeanne Clelland.

Each book covers similar foundational material, but their subsequent emphases are markedly different. For Ivey and Landsberg an important goal is the application of Cartan moving frames to problems in complex algebraic geometry and the explicit solutions of PDEs via the Darboux method. For Jeanne Clelland, the overall purpose is the computation of local geometric invariants for curves and surfaces in various 3-dimensional homogeneous spaces (Euclidean, Minkowski, Equi-affine and projective). This material is covered in the five central chapters of her book, which concludes with chapters on further aspects of differential geometry and non-homogeneous situations, such as moving frames on Riemannian manifolds.

Part 1(Background Material) introduces some profound concepts and powerful techniques, which are subsequently re-examined in a good variety of (low-dimensional) geometric contexts. An essential idea is, of course, that of homogeneous space (in which, I think, ‘all points are the same’). This notion emerges from the intuitive setting of Euclidean space and becomes generally defined as the set \(G/H\) of left cosets of a closed subgroup \(H\) of a Lie group \(G\). Frame bundles and Maurer-Cartan forms are the other main components underlying the method of moving frames.

The most rewarding aspect of this book (Part 2) relates to the outcome of its main aim — the exploration of local geometric invariants for curves and surfaces in submanifolds of various 3-dimensional homogeneous spaces. For curves in Euclidean space, curvature and torsion are unchanged by rigid motions, but these aren’t necessarily preserved by equi-affine transformations \(T:\mathbb{R}^3\to\mathbb{R}^3\). Arc-length is another matter in which interesting comparisons can be made. For curves \(\alpha:I\to\mathbb{R}^3\) it is defined in terms of the first derivative \(\alpha'\), but the equi-affine arc-length of a curve \(\alpha:I\to A^3\) is given in terms of \(\alpha'\), \(\alpha''\), and \(\alpha'''\). In Minkowski space, curves parametrized by proper time differ by a Lorentz transformation if, and only if, they have the same Minkowski curvature and Minkowski torsion. Indeed, this book provides an unusual mathematical basis for the understanding of special relativity.

Apart its mathematical content, another fine feature of this book is the clarity of the (almost conversational style) of Jeanne Clelland’s narrative. Readers always have a clear sense of the route being taken and there are frequent verbal recaps of previously discussed ideas. Also, instead of appearing at the end of chapters, the exercises are dispersed through the text. Many of these invoke the use of the computer algebra system Maple^{TM}.

Primarily intended for ‘beginning graduate students’, this book is highly recommended to anyone seeking to extend their knowledge of differential geometry beyond the undergraduate level.

Peter Ruane is retired from the field of mathematics education, which involved the training of primary and secondary school teachers. His postgraduate study included of algebraic topology and differential geometry, with applications to superconductivity.