An Introductory Course on Differentiable Manifolds

This is a new introduction to differentiable manifolds from Dover’s Aurora series of Modern Math Originals. It is aimed at advanced undergraduates and first year graduate students. Because of the prerequisites (“rigorous multivariable calculus, linear algebra, elementary abstract algebra and point set topology”) and the level of sophistication, it is probably best suited to graduate students.

The author takes a rather unusual approach. He has chosen to work from small scale to large scale, so he starts with point-wise matters in Part I, moves to local structures in Part II, advances to global constructions in Part III, and finally reaches geometric structures in Part IV. The author indicates in his preface that he has recently begun to teach his course this way, and he feels that it works. He argues that by establishing algebraic and computational skills first and focusing on local structures, he prepares the student better to appreciate the value of global structures.

“Point-wise”, to the author, means the tensor algebra of linear spaces and their mappings. In this he is thinking specifically of what happens at a single tangent space on a manifold independent of what occurs nearby. Then, “local” takes into account neighboring points, brings in the derivative, and develops integration of vector fields. The main body of the book then addresses “global” with the introduction of manifolds and a limited amount of vector bundle theory. Finally, “geometric structures” includes a single chapter that introduces connections on the tangent bundle as well as the covariant derivative, curvature and torsion.

Besides the pedagogical reasons he offers and the logical flow that his approach suggests, the author also finds historical justification for his treatment. He notes that the pioneers of differential geometry used coordinates and local computations with tensors before any formal definition of a manifold appeared.

Whatever the justification, new students might feel considerable bewilderment when they encounter an abstract treatment of tensor algebra for the first time without some justification or sense of where it was leading. Surely students would benefit from at least a preliminary peek at the big picture before digging into multilinear algebra.

The author is very thorough and he leaves out very few details in development and proofs of basic results. He understands that students often need reminders of previous results, and he is generous in providing them. More examples and more figures would be desirable, but the ones that are provided are well chosen. The exercises are plentiful. Most of them ask for proofs or examples.

Bill Satzer (bsatzer@gmail.com) was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.