geodesics in differential geometry

Question

Let gamma be a straight line in a surface M. How can we prove that gamma is a geodesic?

ALl I note is that a geodesic on a surface M is a unit speed curve on M with geodesic curvature = 0 everywhere.

Update: To making it not look like the question is a tautology, check out this:

http://www.physicsforums.com/showthread.php?t=407105

I'm trying to fill in the gaps and understand the argument for proving this theorem. Thanks

Answer 1

Use the formula $$\kappa^2 = \kappa_g^2 + \kappa_n^2.$$

Here, $\kappa_g$ is the geodesic curvature, $\kappa_n$ is the normal curvature, and $\kappa$ is (unfortunately) just called the "curvature" (it is the $\kappa$ that appears in the Frenet-Serret Formulas).

A straight line has $\kappa = 0$, so....