This book is an elementary account of the geometry of curves and surfaces. It is written for students who have completed standard courses in calculus and linear algebra, and its aim is to introduce some of the main ideas of differential geometry.
The language of the book is established in Chapter 1 by a review of the core content of differential calculus, emphasizing linearity.
Chapter 2 describes the method of moving frames, which is introduced, as in elementary calculus, to study curves in space. (This method turns out to apply with equal efficiency to surfaces.) Chapter 3 investigates the rigid motions of space, in terms of which congruence of curves and surfaces is defined in the same way as congruence of triangles in the plane.
Chapter 4 requires special comment. One weakness of classical differential geometry was its lack of any adequate definition of surface. In this chapter we decide just what a surface is, and show that every surface has a differential and integral calculus of its own, strictly analogous to the familiar calculus of the plane. This exposition provides an introduction to the notion of differentiable manifold, which is the foundation for those branches of mathematics and its applications that are based on the calculus.
The next two chapters are devoted to the geometry of surfaces in 3-space. Chapter 5 measures the shape of a surface and derives basic geometric invariants, notably Gaussian curvature. Intuitive and computational aspects are stressed to give geometrical meaning to the theory in Chapter 6.
In the final two chapters, although our methods are unchanged, there is a radical shift of viewpoint. Roughly speaking, we study the geometry of a surface as seen by its inhabitants, with no assumption that the surface can be found in ordinary three-dimensional space. Chapter 7 is dominated by curvature and culminates in the Gauss-Bonnet theorem and some of its geometric and topological consequences. In particular we use the Gauss-Bonnet theorem to prove the Poincaré-Hopf theorem, which relates the singularities of a vector field on M to the topology of M.
Chapter 8 studies local and global properties of geodesics. Full development of the global properties requires the notion of covering surface. With it, we can give a comprehensive survey of surfaces of constant curvature and prove the theorems of Bonnet and Hadamard on, respectively, positive and nonpositive curvature.
No branch of mathematics makes a more direct appeal to the intuition than geometry. I have sought to emphasize this by a large number of illustrations, which form an integral part of the text.
Each chapter of the book is divided into sections, and in each section a single sequence of numbers designates collectively the theorems, lemmas, examples, and so on. Each section ends with set of exercises; these range from routine tests of comprehension to more challenging problems.
In this revision. the structure of the text, including the numbering of its contents, remains essentially the same, but there are many changes round this framework. The most significant are, first, correction of all known errors; second, a better way of referencing exercises (the most common reference); third, a general improvement of the exercises. These improvements include deletion of a few unreasonably difficult exercises, simplification of others, and fuller answers to odd-numbered ones.
In teaching from the first edition of this book I have usually covered the background material in Chapter 1 rather rapidly and not devoted any classroom time to Chapter 3. A short course in the geometry of curve and surfaces in 3-space might consist of: Chapter 2 (omit Sec.8), Chapter 4 (omit Sec.8), Chapter 5, Chapter 6 (covering Secs.6-9 lightly), and a leap to Section 6 of Chapter 7: the Gauss-Bonnet theorem. This is essentially the content of a traditional undergraduate course in differential geometry, with clarification of the notions of surface and mapping.
Such a course, however, neglects the shift of viewpoint mentioned above In which the geometric concept of surface evolved from a shape in 3-space to an independent entity---a two-dimensional Riemannian manifold.
This development is important from a practical viewpoint since it makes surface theory applicable throughout the range of scientific applications where 2-parameter objects appear that meet the requisite conditions---for example, in the four-dimension manifolds of general relativity.
Such a surface is logically simpler than a surface in 3-space since it is constructed (at the start of Chapter 7) by discarding effects of Euclidean space. However, readers can neglect this transition and proceed directly to most of the topics considered in the final two chapter, for example, properties of geodesics (length-minimization and completeness), singularities of vector fields, and the theorems of Bonnet and Hadamard.
For readers with access to a computer containing Mathematica, Maple, or other symbolic computation programs, I have included some forty computer exercises. These afford an opportunity to amplify the text in various ways.
Previous computer experience is not required. The Appendix contains a summary of the syntaxes of recent versions of Mathematica and Maple, together with a list of explicit computer commands covering the basic geometry of curves and surfaces. Further commands appear in the answers to exercises.
It is important to go, step by step, through the hand calculation of the Gaussian curvature of a parametrized surface, but once this is understood, repetition becomes tedious. A surface in R3 given only by a formula is seldom easy to sketch. But using computer commands, a picture of a surface can be drawn and its curvature computed, often in no more than a few seconds. Analogous remarks hold for space curves.
Among other applications appearing in the exercises, the most valuable, since unreachable by humans, is the numerical solution of differential equations---and the plotting of these solutions.
This book would not have been possible without generous contributions by Allen B. Altman and Joseph E. Borzellino.