1. Let σ be the ruled surface generated by the binormals b of a unit-speed curve γ:
σ(u, v) = γ(u) + vb(u).
Show that the first fundamental form of σ is
(1 + v2τ 2) du2 + dv2,
where τ is the torsion of γ.
2. Let f be a smooth function and let
σ(u, v) = (u cos v, u sin v, f(u))
be the surface obtained by rotating the curve z = f(x) in the xz-plane around
the z-axis.
(i) Find all functions f for which σ is conformal.
(ii) Find all functions f for which the surface patch σ.