Curvature of Surfaces

Abstract

Let S be an n-surfacc in ℝ^{n +1}, oriented by the unit normal vcctor field N, and let p ∈ S. The Weingarten map L _p : S _p → S _p , defined by L _p (v) = − ∇_v N for v ∈ S _p , measures the turning of the normal as one moves in S through p with various velocities v. Thus L _p measures the way S curves in ℝ^{n +1} at p. For n = 1, we have seen that L _p is just multiplication by a number K(p) the curvature of S at p. We shall now analyze L _p when n > 1.