Geodesics on spheroid

Question

Describe the geodesics

A Spheroid obtained by rotating the ellipse $\frac{x^2}{p^2}+\frac{z^2}{q^2}=1$ around the z-axis where $p, q\gt 0$

Please explain this question explicitly. Thank you:)

Answer 1

First of all you need to know what is the connection you want to compute the geodesics of. If you want a metric connection you must know the metric at play, which in most cases will be the one induced by embedding in $R^3$.

Once you know that you can solve the geodesics equation with respect to appropriate coordinates and find an analytic expression for them, or just give that equation as a more implicit answer: depending on which degree of description you seek.

I would suggest trying cylindrical coordinates to begin with, sincr it should be easier to show the required property.

Answer 2

There is an excellent account on this Wikipedia page. The first two-thirds of the article cover ellipsoids of revolution (which is what you have).