Assignment 6, due 6 June - Institut für Mathematik

Technische Universität Berlin
Institut für Mathematik
Pinkall/Lam
Summer 2016
Due: 6 June 2016
http://www3.math.tu-berlin.de/geometrie/Lehre/SS16/DGI
Differentialgeometrie I: Kurven und Flächen
Homework 6
Problem 1
8 points
Let γ = (γ1 , γ2 ) be a regular plane curve. It induces a surface of revolution σ(u, θ) =
(γ1 (u) cos(θ), γ1 (u) sin(θ), γ2 (u)).
(a) Determine the condition on γ such that σ is regular. Sketch an example where
σ fails to be regular.
(b) Reparametrize γ such that σ is conformal.
Problem 2
Let F : R3 → R so that gradp F 6= 0 for all p ∈ F −1 ({0}). Show that
8 points
(a) At every q ∈ F −1 ({0}) there is an open neighborhood U ⊂ R3 of q such that
U ∩ F −1 ({0}) is a surface parametrized by some map f .
(b) The parametrized surface f can be chosen such that the normal vector field is
N = (| grad F |−1 grad F ) ◦ f .
(c) Let F (x, y, z) = z 2 . Show that grad(0,0,0) F = 0 and F −1 ({0}) is a regular surface
containing (0, 0, 0).
p
(d) Let F (x, y, z) = z 2 + ( x2 + y 2 − R)2 − r2 , 0 < r < R. Sketch F −1 ({0}) and
explicitly find a regular parametrization of F −1 ({0}).

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