Chapter 1 - The Mathematics of Curvature

Abstract

This chapter reviews some of the mathematical tools that are required to describe special properties of curved surfaces. The tools are to be found in differential geometry, analytical function theory, and topology. A particular class of saddleshaped (hyperbolic) surfaces called “minimal surfaces” is treated with special attention because they are relatively straightforward to treat mathematically and do form good approximate representations of actual physical and chemical structures. The principal curvatures can be combined to give two useful measures of the curvature of the surface: the Gaussian curvature (K) and the mean curvature (H). The Gaussian curvature has a number of interesting geometrical interpretations. One of the more striking is connected with the Gauss map of a surface, which maps the surface onto the unit sphere. The Gauss-Bonnet theorem is a profound theorem of differential geometry, linking global and local geometry. A minimal surface can be represented (locally) by a set of three integrals. They represent the inverse of a mapping from the minimal surface to a Riemann surface. The P-surface, the D-surface, and the gyroid are the simplest members of a large family of structures whose members are still being identified. In many ways these three surfaces are the most important; they have been identified in a variety of physical systems, from silicates to cells.