The eminently descriptive back cover description of the contents of Jeffrey M. Lee’s Manifolds and Differential Geometry states that “[t]his book is a graduate-level introduction to the tools and structures of modern differential geometry [including] topics usually found in a course on differentiable manifolds, such as vector bundles, tensors, differential forms, de Rham cohomology, the Frobenius theorem, and basic Lie theory.” Additionally, Lee discusses connections on vector bundles, Riemannian manifolds and Lorentz manifolds. Beyond this, the following is, I think, particularly evocative: “There is also a section that derives the exterior calculus version of Maxwell’s equations.” Again, this back-cover blurb is a very good, i.e. representative and rather extensive, description of what is to be found in the roughly 670 pages of the book.
The book is beautifully written, I think: Lee is very clear and very thorough, he gives good examples at the right frequency, his interspersed remarks are apt, and his exercises (peppering the text proper) and problems (coming at chapters’ ends) are quite good. For a variety of reasons, not least of which being the recent triumph of Perelman over Poincaré-3, i.e. the exploitation of Hamilton’s Ricci flow, and, before that, and closely connected to it, Thurston’s geometrization conjecture, there are many books around these days on the subject of manifolds and differential geometry. I think that Lee’s book compares very well with these.
On a personal note, I have reviewed a number of books in this area in this column, among them Loring Tu’s An Introduction to Manifolds and A. L. Besse’s Einstein Manifolds. I would class the book under review as a mean between these two extremes in the sense that the indicated sequence would make for a nice two or three year course of studies leading to some rather avant garde mathematics (at the research level). However, the mean does not uniquely determine its extremes, so other sources are certainly available, e.g., respectively, Michael Spivak’s Calculus on Manifolds and his titanic Differential Geometry (or am I dating myself here?). In any case, there are many options.
Additionally, differential geometry is of course ascendant nowadays due to the renewed friendship between physics and geometry, or in fact mathematics itself. It is very nice, therefore, to get in Lee’s book a treatment of the Maxwell equations in the setting of the exterior calculus, as well.
All this being said, Manifolds and Differential Geometry is poised to be a major player as far as introductory level graduate text books in differential geometry are concerned. It is a fine book.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.