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 $\mathbb R^4$.

  • $\begingroup$ Thanks this occured to me but can there be such an example in R3? $\endgroup$ – kroner Feb 14 '16 at 0:47
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    $\begingroup$ @zbigniew2015: If all you want is a closed geodesic (and you don't care whether the entire manifold is compact), take a straight infinite cylinder with a curve going perpendicularly around it. $\endgroup$ – Henning Makholm Feb 14 '16 at 0:56

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