Differential Geometry and Lie Groups: A Computational Perspective

The book under review is the first of a two-volume sequence on differential and Riemannian geometry which is intended "for a wide audience ranging from upper undergraduate to advanced graduate students in mathematics, physics, and more broadly engineering students, especially in computer science."  The totality of contents over both volumes covers the standard topics for a one-year course sequence in differential geometry at the graduate level.

 

The core of this first volume shares contents with standard Riemannian geometry texts but includes much more preparatory material and emphasizes examples and applications to Lie groups. The authors have deferred the algebra of differential forms and the full bundle formalism to the second volume. Thus, connections and curvature are presented here only on the tangent bundle. Other common topics in differential geometry deferred to the second volume include distributions and the Frobenius Theorem, integration of forms, and Hodge theory. In exchange there are several less common topics such as Lorentz and O(p,q) groups, constructing manifolds from gluing data, and the substantial discussion of homogeneous and symmetric spaces.

 

The text is divided into two parts, but for this review we further divide each into two units. The first of these four units is focused on the classical groups and group actions, including discussion of topological groups, and Grassmannians and Stiefel manifolds. The exposition is concrete and well-paced for students. There are many color illustrations and substantial end-of-chapter problems, not just problems asking for proofs of results from the text.

 

The text then turns to basic manifold theory, first studying the case of the classical groups as embedded submanifolds before developing the abstract formalism. Especially helpful for students is a detailed discussion of various definitions for a tangent vector and the subtleties of constructing manifolds from gluing data. Tangent and cotangent bundles, vector fields, flows, and Lie derivatives are all developed. Incidentally, the text considers \( C^{k} \)-differentiable manifolds rather than assuming infinitely-smooth manifolds as do some other authors.

 

The third unit moves onto the core topics of Riemannian geometry: connections and curvature, geodesics and the exponential map, first and second variation formulas, Jacobi fields and conjugate points. The order of topics is conventional and some of the technical proofs are omitted with reference given or assigned as problems for the reader. Their approach is inspired by the excellent exposition in Milnor's Morse Theory, as they acknowledge in the preface. Curiously, there is no discussion of Riemannian submanifolds.

 

Finally, in the fourth unit the machinery thus developed is applied to Lie groups. Other introductory texts also discuss applications to Lie groups, but they often distribute this material throughout their exposition of the various concepts. The coverage here is stronger than in many other introductory texts, including an extended discussion of homogeneous spaces and symmetric spaces. There are again many substantial multi-part problems of benefit to students.

 

Sandwiched between the second and third units are two chapters on basic analysis and general topology, respectively. They are designed as a refresher, being likely too condensed as a first encounter for many students.

 

In summary, the authors have succeeded in writing a distinctive introductory differential geometry text by using the classical groups for early motivation and then extensively illustrating the developed abstraction in the setting of Lie groups and homogeneous spaces. The concreteness of the former would be particularly welcomed by readers from applied disciplines. The text’s coverage is extensive, its exposition clear throughout, and the color illustrations helpful. The authors are also familiar with many texts at a comparable level and have drawn on them in several places to include some of the most insightful proofs already in the literature.

 

A comment about the book's intended audience: though ostensibly addressed also to those in applied disciplines, such readers may find the style and pace in the latter half of the book challenging. This is a book that is suitable for use in a math program and makes attendant demands of the reader. Also, differential geometry texts written with an eye for applied disciplines usually have chapters devoted to specific applications, e.g. electromagnetism, classical mechanics, fluid mechanics, relativity, etc. Such is curiously missing from the text under review. Occasionally, applications are mentioned in passing or a reference is given, but not having them worked out within the text may limit the book's appeal to those in applied disciplines.

 

Nonetheless, the text admirably guides the reader along a fascinating journey from the concrete matrix exponential of its opening pages to the worlds of homogeneous and symmetric spaces of its final chapter.

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Jer-Chin Chuang is an instructor in mathematics at the University of Illinois Urbana-Champaign.