Assume a two dimensional surface $\Sigma$ parametrized by $f:U\rightarrow\Sigma$, where $U \subset \mathbb{R}^2$, $ \Sigma \subset \mathbb{R}^3$. Show that if for every $a,b \in \mathbb{R^2}$, $\gamma(t)=f(at+b)$ is a geodesic line then $\Sigma$ is isometric to a part of a plane.
I have tried working with geodesic equations but no luck so far. I feel like i lack some basic understanding, so references to good notes on the subject will be appreciated as well.
Thanks!