Shape and Curvature

In this chapter, we study the relationship between the geometry of a regular surface M in 3-dimensional space ℝ³ and the geometry of ℝ³ itself. The basic tool is the shape operator defined in Section 13.1. The shape operator at a point p of M is a self-adjoint linear transformation S of the tangent space M _p that measures how M bends in different directions. The shape operator can also be considered to be (minus) the differential of the Gauss map of M (Proposition 13.5), and its effect is illustrated using tangent vectors to coordinate curves on the surface and the image of these vectors on the sphere (Figure 13.1).