DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES 5. The Second Fundamental Form of a Surface

The main idea of this chapter is to try to measure to which extent a surface S is different from a plane, in other words, how “curved” is a surface. The idea of doing this is by assigning to each point P on S a unit normal vector N(P ) (that is, a vector perpendicular to the tangent plane at P ). We are measuring to which extent is the map from S to R given by P 7→ N(P ) (called the Gauss map) different from the constant map, so we are interested in its derivative (or rather, differential). This will lead us to the concept of second fundamental form, which is a quadratic form associated to S at the point P . 5.1. Orientability and the Gauss map. Let S be a regular surface in R.