TITLE: Geodesic of a curved surface
QUESTION [12 upvotes]: I'm trying to read Lambourne's Relativity, Gravitation and Cosmology, but as this seems more of a maths question I've posted it here rather than in the physics forum.
The author talks about affinely parameterized geodesics and then, in Exercise 3.9, shows that the equator on the surface of a sphere is a geodesic.
My question is this: 
How do you find the geodesic of a simple curved surface that isn't a sphere (for example, $z = x^2 + y^2$)?
I'm guessing that because the geodesic equation contains Christoffel symbols I would first need to find the metric for that particular surface, but I don't know how to do that.
I'm therefore guessing that my second question would need to be how do you find the metric for a simple curved surface (again for example $z = x^2 + y^2$).
Apologies if I've asked anything ridiculous.
Thank you.

REPLY [3 votes]: How about this: for $z=F(x,y)$, a curve is given by 
$$              X(t)= ( x(t), y(t), F(x(t),y(t)) )$$
The surface normal is 
$$               N = \left( \frac{dF}{dx}, \frac{dF}{dy}, -1 \right)$$
Want a unit speed curve:
$$         |X'|^2=1$$
Want curve acceleration parallel to the normal $               ( \frac{dF}{dx}, \frac{dF}{dy}, -1 )$, so
$$    \operatorname{cross}( X'', N )=0$$
That get's messy quickly - four non-linear PDE's. But it is fun to play with eg $ z=x^2+y^2$