Curves and Surfaces
At present, quite a few textbooks are available describing the classical theory of curves and surfaces in Euclidean three-dimensional space. This book is special in the sense that together with the standard local theory, a lot of attention is paid to global problems. The first three chapters of the books contain a description of the local theory for curves and surfaces. Chapter 4 is devoted to the topology of the space. A higher dimensional version of the well known Jordan curve theorem is discussed. The proof is based on a version of degree theory. Existence of a tubular neighbourhood is used here as well as in chapter 5, where surface integration is treated. This chapter also includes the Gauss theorem and the Brower fixed point theorem. Gauss’ Theorema Egregium is explained in chapter 7. Global geometry of surfaces with a positive Gauss curvature, the Alexandrov theorem and the isoperimetric inequality are described in chapter 6. The book culminates with the Gauss-Bonnet theorem (chapter 8). The last chapter is then devoted to the global geometry of curves. Each chapter ends with a set of exercises and hints for their solutions. As a prerequisite for reading the book, a few basic facts from linear algebra, topology and ordinary differential equations are needed, together with Lebesgue integration. This textbook is well organized and nicely written. It is suitable for self-study and also for teachers preparing courses on the theory of curves and surfaces.