DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

§2. The Gauss Map and the Second Fundamental Form 51EXERCISES 2.2*1. Check that the parameter curves are lines of curvature if and only if F = m =0. Show,moreover, that in this event, we have the principal curvatures k 1 = l/E and k 2 = n/G.♯ 2. a. Show that the matrix representing the linear map S P : T P M → T P M with respect to thebasis {x u , x v } isI −1P II P =[EF] −1 [F lG m]m.n(Hint: It suffices to check S P (x u ) · x u , S P (x u ) · x v , and so on.)ln − m2b. Deduce that K =EG − F 2 .3. Compute the second fundamental form II P of the following parametrized surfaces. Then calculatethe matrix of the shape operator, determine H and K.a. the cylinder: x(u, v) =(a cos u, a sin u, v)*b. the torus: x(u, v) =((a + b cos u) cos v, (a + b cos u) sin v, b sin u) (0