I was asked this in differential geometry class and it.is bugging me as i do not know the answer I know a surface of constant Gaussian curvature zero is locally isometric to a plane on which no geodesic is closed as they are lines but the fact that it is a local isometry puzzles me Does someone have an example of a surface with zero Gaussian curvature and a closed geodesic on it? Thanks all
The most basic example is the "flat torus": a rectangle where you glue the top and bottom sides together, and then also glue the left and right sides together.
Then, for example, a vertical line on the rectangle is a closed geodesic that goes around the torus.
The flat torus doesn't embed smoothly into $\mathbb R^3$, but it does in $\mathbb R^4$.