Differential Geometry and Its Applications

John Oprea begins Differential Geometry and Its Applications with the notion that differential geometry is the natural next course in the undergraduate mathematics sequence after linear algebra. He argues that once students have studied some multivariable calculus and linear algebra, a differential geometry course provides an attractive transition to more advanced abstract or applied courses. His thoughtful presentation in this book makes an excellent case for this. As he says, the natural progression of concepts in differential geometry allows the student to progress gradually from calculator to thinker.

This edition of the text is over a hundred pages longer than the first edition. Evidently Oprea has incorporated many suggestions from those who have taught from the text. There is a good deal to like about this book: the writing is lucid, drawings and diagrams are plentiful and carefully done, and the author conveys a contagious sense of enthusiasm for his subject.

Most of the text concentrates on differential geometry in three dimensions, and much of it focuses on the usual topics of curves and surfaces. The final chapter provides a quick look at higher dimensions, for which the author has laid a very adequate foundation. (For example, the students will already have seen the covariant derivative, shape operator, parallel transport, and the Christoffel symbols.) The text makes extensive use of Maple (specifically Maple 10) as a computer algebra system. (Mathematica could easily be substituted for Maple throughout.) Using the computer algebra system enables a variety of interesting but algebraically complex examples throughout, such as those involving elliptic functions. Enhanced capabilities for visualization are also obvious benefits of using such a software package. The author provides, for example, a Maple routine for plotting geodesics on surfaces and this becomes a central part of his treatment of this subject.

A notable characteristic of Oprea’s approach is the way he establishes and maintains connections with other areas of mathematics — complex analysis, differential equations, and the calculus of variations, for example — as well as with a variety of applications. (These are mostly applications to physics.) One notable engineering application is a design for the shoulder of a packaging machine wherein a developable surface is required to avoid stretching or tearing the packaging. While not a glamorous example, this demonstrates the power of differential geometry in a very practical situation.

The chapter on holonomy and the Gauss-Bonnet theorem is typical of the author’s approach. He discusses the question of what parallel transport of vectors might mean, introduces the concept of holonomy, and provides an example using Foucault’s pendulum. Then he proves the Gauss-Bonnet theorem and offers several examples showing how it can be used.

This is a very attractive textbook for a first course in differential geometry and one well worth consideration. The author says that a bare minimum background is first-year calculus, but at least of bit of multivariable calculus, some linear algebra and maybe a sprinkling of differential equations would be most helpful.

Bill Satzer (wjsatzer@mmm.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.