The question is as follows. Imagine we have two closed geodesics on a surface, where these two geodesics bound a cylinder together and the geodesics are diffeomorphic to circles. Can the cylinder have $K>0$ everywhere or $K<0$ everywhere? Here $K$ is Gaussian Curvature.
What's confusing me about this question is that a cylinder has Gaussian Curvature of $K=0$ everywhere, so wouldn't the answer to both questions be no? But then I don't get the point of the question with the geodesics.
Thank you in advance
The point is that you don't know the metric on the surface in advance. Consider for example the torus of revolution with the metric induced by the Euclidean one and take two meridians together with the cylinder bounded by them. The meridians are circles and geodesics but this cylinder doesn't have $K = 0$. It has areas in which $K > 0$ and areas in which $K < 0$.
In general, if you apply the Gauss-Bonnet theorem to your cylinder $C$, you'll get
$$ \int_{C} K \, dA = 2\pi \chi(C) = 0 $$
so you can't have $K > 0$ everywhere or $K < 0$ everywhere.