This is a text on the classical differential geometry of curves and surfaces from a modern point of view. In recent years, there has been a marked increase in the interest in the global aspects of this topic, which is the emphasis here. In order to study the global properties of curves and surfaces, more sophisticated tools are necessary than are usually found in texts on the same topic. Also, topological questions arise and must be understood. Thus, for instance, there is a treatment of the Gauss-Bonnet theorem and a discussion of the Euler characteristic. The authors also cover Alexandrov's theorem on compact surfaces in $\mathbb{R}^3$ with constant mean curvature. The last chapter discusses the global geometry of curves, including periodic space curves and the four-vertices theorem in the general case where the curves are not necessarily convex.

Besides being an introduction to the lively subject of curves and surfaces, this book can also be used as an entry to a wider study of differential geometry in general. It is suitable for a first-year graduate course or an advanced undergraduate course.

Readership

Advanced undergraduates, graduate students, and research mathematicians interested in the differential geometry of curves and surfaces.