Assignment 10, due 4 July - Institut für Mathematik

Summer 2016
Technische Universität Berlin
Institut für Mathematik
Pinkall/Lam
Due: 4 July 2016
http://www3.math.tu-berlin.de/geometrie/Lehre/SS16/DGI
Differentialgeometrie I: Kurven und Flächen
Homework 10
Problem 1
Find the asymptotic lines of the surfaces defined by z = xy and z =
4 points
x
y
+
y
.
x
Problem 2
4 points
Show that the following surfaces share the same Gaussian curvature but their first
fundamental forms are different:
f (u, v) = (u cos v, u sin v, log u),
g(u, v) = (u cos v, u sin v, v).
Problem 3
4 points
Let γ̂ : [a, b] → M be a curve on a regular surface g : M → R3 . We denote γ := g ◦ γ̂.
Show that the following are equivalent:
a) γ̂ is a curvature line.
b) The surface normal ν of g defines a regular surface f (x, y) := γ(x) + yν(γ̂(x)),
which is developable, i.e. K = 0.
Problem 4
4 points
Show that if (the image of) an asymptotic line of a regular surface with K < 0
is a Frenet curve with torsion τ , then the mean curvature of the surface satisfies
H = ±τ cot φ, where φ is the angle between the two asymptotic lines.

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