DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

44 Chapter 2. Surfaces: Local TheoryFigure 2.2yet again, we have:(†)±κ = κN · n = T ′ (0) · n = −T · (n◦α) ′ (0) = −D V n(P ) · V.This leads us to study the directional derivative D V n(P ) more carefully.Proposition 2.1. For any V ∈ T P M, the directional derivative D V n(P ) ∈ T P M. Moreover,the linear map S P : T P M → T P M defined byS P (V) =−D V n(P )is a symmetric linear map; i.e., for any U, V ∈ T P M,wehave(∗)S P (U) · V = U · S P (V)S P is called the shape operator at P .Proof. Forany curve α: (−ε, ε) → M with α(0) = P and α ′ (0) = V, weobserve that n◦α hasconstant length 1. Thus, by Lemma 2.1 of Chapter 1, D V n(P )·n(P )=(n◦α) ′ (0)·(n◦α)(0) = 0, soD V n(P )isinthe tangent plane to M at P . That S P is a linear map is an immediate consequenceof Proposition 2.3 of the Appendix.Symmetry is our first important application of the equality of mixed partial derivatives. Firstwe verify (∗) when U = x u , V = x v . Note that n · x v =0,so0= ( n · x v)u = n u · x v + n · x vu .(Remember that we’re writing n u for D xu n.) Thus,S P (x u ) · x v = −D xu n(P ) · x v = −n u · x v = n · x vu= n · x uv = −n v · x u = −D xv n(P ) · x u = S P (x v ) · x u .