Assume that E = 1, F = M = 0, where E, F, G, L, M, N denote coefficients of the first and the second fundamental forms of a smooth surface.

(a) (5 marks) Prove that L = L(u, v) does not depend of v.

(b) (5 marks) Prove that if G(u, v) = u 2 , then at every point either L = 0 or N = 0.

(c) (5 marks) Prove that if G = u 2 and L = 0, then N(u, v) = c(v)u, where c(v) is function that depends only on v.

(d) (5 marks) Prove that if G = u 2 , then the surface is locally isometric to the plane.

Assume that E = 1, F = M = 0, where E, F, G, L, M, N denote coefficients of the first and the second fundamental forms of a smooth surface. (a) Prove that L = L(u, v) does not depend of v. (b) Prove that if G(u, v) = us, then at every point either L = 0 or N = 0. (c) Prove that if G = u2 and L = 0, then N(u, v) = c(v)u, where c(v) is function that depends only on v. (d) Prove that if G = u, then the surface is locally isometric to the plane