Elementary Differential Geometry

Differential geometry represents an elegant collision of geometry, analysis and algebra with obvious broad applications in many areas of mathematics as well as in fields as diverse as molecular biology and structural mechanics. A difficult question it raises, however, is how to introduce the subject to students. I am aware of two very distinct approaches. The first, represented by Barrett O’Neill’s Elementary Differential Geometry, reaches for the full formalism with differential forms, connections, and covariant derivatives. On the other end, there are books like Pressley’s Elementary Differential Geometry that essentially avoid the formalism completely. The former treatment can easily overwhelm students, while the latter can create a big discontinuity for students who move on to more advanced courses in differential geometry.

The current book attempts to split the difference. The author suggests that he avoids formalism “as much as possible”. He concentrates on problems of broad geometrical interest pursued using fundamental tools from analysis and linear algebra. Formalism is downplayed, but it is not invisible. So we see differentials, but differential forms are never mentioned. Local coordinates are used routinely, but covariant derivatives are introduced midway through the book and are used fairly extensively thereafter.

The study of differential geometry begins in the second chapter with an extensive discussion of curves, primarily in two and three dimensions. (The first chapter does formal Euclidean geometry based on Hilbert’s axioms. It is largely independent of the rest of the book.) The following chapter then takes up classical surface theory. Here, as in the rest of the book, the author does an excellent job of integrating geometry with analysis and linear algebra. He provides many good examples and exercises throughout, and his proofs are clear and detailed. I particularly liked his presentation of Fenchel’s theorem (how much does a space curve need to curve to be able to close up), the Fáry-Milnor theorem (how much a curve needs to curve in order to be knotted) as well as the four-vertex theorem.

A very meaty fourth chapter brings us to intrinsic differential geometry. This introduces students to the covariant derivative, the Riemannian curvature tensor, Riemannian metrics, geodesics, and parallel transport. Here, without the formalism, it’s easy to lose sight of the forest for the trees. Nonetheless, the author handles the material gracefully, and grinds out one computation after another in local coordinates.

The final two chapters culminate in the Gauss-Bonnet theorem. The author first proves Gauss’ divergence theorem and uses it to show that the total Gaussian curvature of a closed surface does not depend on the Riemannian metric. Then the final chapter concentrates on the topological interpretation of the total Gaussian curvature. The author proves that every compact surface can be triangulated, and uses that to prove the Gauss-Bonnet theorem.

This is an attractive book, one that would be appealing for an introductory course.

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.