DIFFERENTIAL GEOMETRY

Perugia Aug 07

Assignment Week 3

(1) Write up some of the problems on the exam that you were not able to do or only do partially and hand them in in class on Monday.

(2) Let a metric be given by dx^{2} + f^{2}(x, y)dy^{2}. Show that the curves y =constant are geodesics, in fact minimizing geodesics. What does this say about surfaces of revolution?

(3) Let γ be a curve on a surface parametrized by arc length, which is fixed by an isometry A, i.e. A(γ(t)) = γ(t) for all t. If furthermore A ≠ id, show that γ must be a geodesic by showing that it is locally minimizing. Illustrate with some examples.

(4) (a) Describe the behaviour of all geodesics on the following surfaces of revolution,

where f(s) is the distance to the axis of rotation: (i) f(s) = e^{s} (ii) f(s) = cosh s (iii) f(s) = \frac{1}{1+s^{2}}

(b) In each of the above examples describe all closed geodesics.

(5) (a) Show that the catenoid x(u, v) = (cosh v cos u, cosh v sin u, v) is a minimal

surface.

(b) Show that the helicoid y(u, v) = (sinh v cos u, sinh v sin u, u) is a minimal surface.

(c) Show that (cos t)x + (sin t)y is a one parameter family of minimal surfaces

and that they are all isometric to each other.

(6) DoCarmo p.46 #4 , p. 56 #7,8