Hi All, i have been asked to investigate Geodesics on a Torus and their different properties.

Unfortunately I am getting nowhere after trying for 2 hours. I must be missing something crucial...

Given a parameterization for a torus:

$\displaystyle \gamma(u,v)= \left((a+b\cos u)\cos v,(a+b\cos u)\sin v,b\sin u\right)$

Show that if at some point a geodesic is tangent to the top circle, $\displaystyle \gamma(u=\frac{\pi}{2})$, then it remains on the 'outside half' $\displaystyle (-\frac{\pi}{2}\leq u \leq \frac{\pi}{2})$ of the torus. Show also that it oscillates between the top circle and bottom circle.

There are also some other questions that ask the same thing - ie give you a starting point, and then get you to prove some properties of the set of geodesics that satisfy the initial condition.

I'm really stuck with this.

Information I am trying to use: Geodesics have acceleration normal to the surface at all times.... the coefficients of the first fundamental form etc.

Can anyone give me some help with this question or perhaps give me an idea for a method to nut out these types of questions?

Cheers