DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

46 Chapter 2. Surfaces: Local Theory= l(ac)+m(bc + ad)+n(bd).In the event that {x u , x v } is an orthonormal basis for T P M,wesee that the matrix II P representsthe shape operator S P . But it is not difficult to check (see Exercise 2) that, in general, the matrixof the linear map S P with respect to the basis {x u , x v } is given byI −1P II P =[EF] −1 [F lG m]m.nRemark. We proved in Proposition 2.1 that S P is a symmetric linear map. This means thatits matrix representation with respect to an orthonormal basis (or, more generally, orthogonal basiswith vectors of equal length) will be symmetric: In this case the matrix I P is a scalar multiple ofthe identity matrix and the matrix product remains symmetric.By the Spectral Theorem, Theorem 1.3 of the Appendix, S P has two real eigenvalues, traditionallydenoted k 1 (P ), k 2 (P ).Definition. The eigenvalues of S P are called the principal curvatures of M at P . Correspondingeigenvectors are called principal directions. A curve in M is called a line of curvature if itstangent vector at each point is a principal direction.Recall that it also follows from the Spectral Theorem that the principal directions are orthogonal,so we can always choose an orthonormal basis for T P M consisting of principal directions. Havingdone so, we can then easily determine the curvatures of normal slices in arbitrary directions, asfollows.Proposition 2.3 (Euler’s Formula). Let e 1 , e 2 be unit vectors in the principal directions atP with corresponding principal curvatures k 1 and k 2 . Suppose V = cos θe 1 + sin θe 2 for someθ ∈ [0, 2π), aspictured in Figure 2.3. Then II P (V, V) =k 1 cos 2 θ + k 2 sin 2 θ.e 2Vθ e 1Figure 2.3Proof. This is a straightforward computation: Since S P (e i )=k i e i for i =1, 2, we haveII P (V, V) =S P (V) · V = S P (cos θe 1 + sin θe 2 ) · (cos θe 1 + sin θe 2 )= (cos θk 1 e 1 + sin θk 2 e 2 ) · (cos θe 1 + sin θe 2 )=k 1 cos 2 θ + k 2 sin 2 θ,as required.□