Riemannian Geometry is a modern introduction to the subject intended for use in a graduate course. The author tells us that his goals are to introduce the subject, to present basic techniques and results in a coherent manner with a view toward future work, and to present some results that are interesting in their own right. He suggests furthermore that his book is intended for a broad group of students, and presumably this includes students intending to specialize in areas other than geometry.
The author’s stated prerequisites are an introductory course in differentiable manifolds, construction of the associated tensor bundles, and Stokes’ theorem. To this I would add some Lie theory, comfort with the use of differential forms, and the willingness to grind through long chains of definition-theorem-proof. While this is in many ways an elegant treatment of Riemannian geometry, it is a hard elegance that can be very difficult for a student to penetrate.
The book begins with the definition of a connection, proceeds quickly to parallel translation, geodesics, the torsion and curvature tensors, moving frames and the exponential map… and never slows down. There are very few examples along the way. I found twenty-three drawings throughout the book — I counted as I read because they seemed so sparse. Even the drawings that are present are curiously unhelpful. As I read, I kept thinking about the geometers I had known, and how I had never seen two of them together for more than a few minutes without drawing sketches for one another. Why not be more generous with the drawings and sketches? If our intent is to communicate mathematics, why not use all the tools at our disposal?
After the first section’s rapid introduction, the next section addresses Riemannian curvature and concentrates on Jacobi’s equation and the relationship between curvature and geodesics. The third and fourth sections take up respectively Riemannian volumes and Riemannian coverings. A short fifth section discusses surfaces and includes a statement and proof of the Gauss-Bonnet theorem.
The isoperimetric inequalities for both constant and variable curvature are a focus of the second half of the book. Included here is a brief discussion of Richard Hamilton’s work with curvature flow, a technique that played a large part in the resolution of the Poincaré conjecture. In his discussion of the isoperimetric inequalities, as well as the comparison and finiteness theorems, I think we see the author’s true passions. His writing is most animated when he’s busy estimating things, especially integrals.
Each chapter has a good collection of exercises. Indeed, I found the “Notes and Exercises” sections the most engaging part of the book. The author has included “Hints and Sketches” for the exercises as an appendix. There is an index of authors, which is valuable, and a subject index, which is too sketchy.
This is not a book I’d recommend for self-study. As a classroom text, it has promise as long as it is accompanied by a good number of examples, calculations and pictures.
Bill Satzer (email@example.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.