The Gauss Map

Abstract

An oriented n-surface in ℝ^{n +l} is more than just an n-surface S, it is an n-surface S together with a smooth unit normal vector field N on S. The function N:S → ℝ^{n +1} associated with the vector field N by N(p) = (p, N(p)), p ∈ S, actually maps S into the unit n-sphere Sⁿ ⊂ ℝ^{n +1} since ∥N(p)∥ = 1 for all p ∈ S. Thus, associated to each oriented n-surface S is a smooth map N: S → Sⁿ. called the Gauss map. N may be thought of as the map which assigns to each point p ∈ S the point in ℝ^{n +1} obtained by “translating” the unit normal vector N(p) to the origin (see Figure 6.1).