Differential Geometry of Curves and Surfaces

The following sections are included:

• Preface
• Contents

Students at the college and university level are familiar with graphs of functions and parametrized curves. In this chapter, we define the curvature of plane curves and space curves, which measures the “amount of bending” of the curve, and introduce its geometric meanings. As applications, the rotation index for closed plane curves and properties of spirals are introduced. While space curves are closely related to surface theory, which is treated in Chapter II, the rotation index for plane curves is also used in the proof of the Gauss-Bonnet theorem for 2-dimensional manifolds in Chapter III.

We introduce parametrizations of surfaces like we did for curves, and define the first and second fundamental forms. Using these, the Gaussian curvature and mean curvature, which do not depend on choice of parametrization, are introduced, and geometric meanings of these curvatures are explained. Moreover, the principal and asymptotic directions of surfaces, which play important roles in analyzing the shapes of surfaces, are introduced. As an application, we introduce the Gauss-Bonnet theorem, which measures the difference between π and the sum of interior angles of a triangle on a surface.

In the previous chapters, we studied curves and surfaces, but looking at surfaces from the viewpoint of manifolds allows us to obtain a broader perspective. In this chapter, for those readers that are familiar with manifolds, we aim at the Hopf theorem (Theorem 17.4) for constant mean curvature surfaces, and continue with further study of surfaces. We start in Section 12 with a review of differential forms, and use them in Sections 13 and 14 to prove the Gauss-Bonnet theorem on 2-dimensional Riemannian manifolds. As an application of that, in Section 15 we prove the index formula for vector fields on compact oriented 2-manifolds and study the index of umbilics on a surface. In Section 16, we show existence of conformal coordinates for surfaces. Then in Section 17, we introduce the Gauss and Codazzi equations, and the fundamental theorem of surface theory, which will be proven in Appendix B.10, and prove the Hopf theorem. In Section 18, we explain the maximal speed descent property (the property of brachistocrones) for cycloids from the viewpoint of Riemannian geometry. Finally in Section 19, we give a proof of existence of geodesic triangulations on surfaces. In this chapter, we assume that the reader is familiar with vector fields on manifolds, differential forms, wedge products and exterior derivatives. (See the references suggested in the text within this chapter.)

The following sections are included:

• A review of calculus
• The fundamental theorems for ordinary differential equations
• Euclidean spaces

The following sections are included:

• Evolutes and the cycloid pendulum
• Convex curves and curves of constant width
• Line integrals and the isoperimetric inequality
• First fundamental forms and maps
• Curvature line coordinates and asymptotic line coordinates
• Surfaces with K = 0
• A relationship between surfaces with constant Gaussian curvature and with constant mean curvature
• Surfaces of revolution of negative constant Gaussian curvature
• Criteria of typical singularities
• Proof of the fundamental theorem for surfaces

Answers and/or hints to the exercises are given here. In the cases where the answers below are not complete, it is recommended to the readers that they work out complete answers for themselves.

The following sections are included:

• Bibliography
• List of Symbols
• Index