Reference no: EM133237612

Question 1: Let R² = {(x, y, z) ∈ R³;z = -1} be identi?ed with the complex plane C by setting (x, y, -1) = x + iy = ζ ∈ C. Let P : C → C be the complex polynomial

P (ζ ) = a₀ζⁿ + a₁ζ^n-1 + · · · +a_n , a ≠ 0, a_i ∈ C, i = 0, . . . , n.

Denote by Π_N the stereographic projection of S² = {(x, y, z) ∈ R³; x² + y² + z³ = 1} from the north pole N = (0, 0, 1) onto R². Prove that
the map F : S² → S² given by

F (p) = Π_N^-1 o P oπ_N (p), if p ∈ S² - {N},

F (N) = N

is differentiable.

Also, let Π_S be the corresponding projection map from the plane z = 1. Calculate the reparametrization map Π^-1Π_S, and express this map in complex analytic terms.

Question 2: Let G : S→S^ be a map between surfaces which preserves lengths along and angles between the coordinate curves on S (coming from a parametrization x→ (u, v)). Show that G preserves length along and angles between arbitrary curves on S.

Question 3: Suppose coordinate curves on a surface (relative to a parametrization x→ (u, v)) form an "orthogonal net" (intersect orthogonally). Write the differential equation for a curve to bisect the angles between the coordinate curves. Express your answer in terms of x→ and its derivatives in u, v and also v and its derivatives in u with the understanding that the curve is given by v = v(u) in the coordinates.