Gaussian Curvature and the Gauss Map

Abstract

We shall now introduce two new measures of the curvature of a surface, called its gaussian and mean curvatures. Although these together contain the same information as the two principal curvatures, they turn out to have greater geometrical significance. The gaussian curvature, in particular, has the remarkable property, established in Chapter 10, that it is unchanged when the surface is bent without stretching, a property that is not shared by the principal curvatures. In the present chapter, we discuss some more elementary properties of the gaussian and mean curvatures, and what a knowledge of them implies about the geometry of the surface.