Curves and Surfaces, published jointly by the American Mathematical Society and Real Sociedad Matemática Española, is a volume in the AMS's Graduate Studies in Mathematics series. It is a revised and updated English version of an earlier book published in Spanish. The authors believe that study of the classical geometry of curves and surfaces should be an essential part of every mathematician's training, and that it is the best way to introduce students of all kinds to differential geometry.
This authors' approach differs in three distinct ways from those of similar texts. First, Lebesgue integration is used extensively as a powerful tool to obtain global geometrical results. Second, topological questions receive a good deal more attention, both on their own merits and as essential steps in reaching geometrical results. (For example, there is a whole chapter on separation and orientability that includes a detailed proof of the Jordan-Brouwer theorem.) Finally, the authors are determined to use coordinate-free language whenever possible to get the clearest possible statements of theorems and clean proofs.
The text is meticulously put together, with a few odd exceptions. (For example, L. E. J. Brouwer's name is consistently misspelled.) In almost all respects, the authors' exposition is elegant. The proofs are detailed, good examples are selected, exercises are ample and hints for solving the exercises are often provided. Prerequisites include the basics of linear algebra, calculus in two and three variables, some knowledge of the topology of Euclidean space, and the rudiments of ordinary differential equations.
Several elements of the authors' treatment are worthy of note. The integral of a function on a compact surface is defined via the integral on any of its tubular neighborhoods of the function obtained by extending the original one as a constant along each normal segment. This bypasses the usual treatment that takes integration in the plane and transposes it to a surface via parameterization.
A chapter on global extrinsic geometry includes proofs of a spatial isoperimetric inequality and Alexandrov's theorem. (Every compact connected surface with constant mean curvature is a sphere.) Succeeding chapters include proofs of Gauss's Theorem Egregium, the Gauss-Bonnet theorem, and the four vertices theorem for plane curves that need not be convex.
With its readable style and the completeness of its exposition, this would be a very good candidate for an introductory graduate course in differential geometry or for self-study.
Bill Satzer (firstname.lastname@example.org) 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.