Summer 2016
Technische Universität Berlin
Institut für Mathematik
Pinkall/Lam
Due: 17 May 2016
http://www3.math.tu-berlin.de/geometrie/Lehre/SS16/DGI
Differentialgeometrie I: Kurven und Flächen
Homework 3
Problem 1
(4 points)
Show that regular homotopy for regular closed curves defines an equivalence relation.
Problem 2
Let
(4 points)
γ(x) := (cos(x), sin(2x)),
δ(x) :=
cos(x)
cos(x)
, sin(x)
1+sin2 (x)
1+sin2 (x)
.
Show that (γ, 2π) and (δ, 2π) are regular homotopic.
Problem 3
(4 points)
2
Let (γ, τ ) be an arc-length parametrized closed curve and A : R → R2 be a Euclidean transformation so that the curve γ̃ := Aγ satisfies γ̃(0) = 0 and γ̃ 0 (0) =
(1, 0).
(a) Determine A in terms of γ(0) and γ 0 (0).
(b) Find a regular homotopy between γ and γ̃.
Problem 4
Consider the functional
(4 points)
Z
Ẽ(γ) :=
L
κ4 v(t)dt
0
for regular curve γ : [0, L] → R with curvature κ and velocity v(t) := |γ 0 (t)|. Derive
the condition on the curvature κ for γ to be a critical point of Ẽ under all variations
with compact support.
2

Assignment 4, due 23 May - Institut für Mathematik