Analysis and Algebra on Differentiable Manifolds: A Workbook for Students and Teachers

This is a collection of solved problems on differentiable manifolds, Lie groups, and Riemannian geometry. It is intended as a supplement rather than as a standalone text. There are plenty of pictures to accompany the problems, including a few in color. The solutions are detailed, with many calculations written out in full, making some of the book's pages look like a mass of symbols.

The problems are complemented by two chapters, at the end of the book, intended to help orient the reader. The first is a kind of "fascicule de resultats": a collection of definitions and theorems. For most users of the book, the main role of this chapter will be to fix notation. Chapter 8, "Some Formulas and Tables," is more interesting. It collects such things as the general formula for stereographic projection, formulas for Lie derivatives and wedge products of differential forms, and lists of the classical groups and their Lie algebras. I find this chapter very useful, both in solving the problems in the book and as a place to find information quickly when Wikipedia is not reachable.

I suspect the book will be particularly useful to instructors who are not specialists in the subject; most of the results and problems here are likely to be familiar to those working in the field. I would certainly want to have this book on hand if I were to teach such a course.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME. He has taught differential geometry several times, but never at this advanced level.