Notations
$M$ will denote a smooth manifold and $\nabla$ an affine connection on it. A smooth curve $\gamma\colon I \to M$ will be called a geodesic if it is $\nabla$-parallel along itself, that is $\nabla_{\dot{\gamma}(t)}\dot{\gamma}=0$ for every $t \in I$. A geodesic will be said to be maximal if every proper extension of it is not a geodesic.
It is easy to find examples of maximal geodesics which do not self-intersect, like
lines in Euclidean plane, or that intersect in infinitely many points, like great circles on the sphere. On the contrary I cannot find examples of geodesics which self-intersect at finitely many points, like the curve below:
Question Is it possible to determine $M$ and $\nabla$ in such a way that one of the resulting maximal geodesics intersects at finitely many points?
Thank you.
Best Answer
Take a quadrant of the plane and roll it up into a cone by gluing the two edges.
Shown above are the unfolding and the glued manifold, with a single geodesic shown in black; the "seam" runs down the right side of the cone in the right image. In fact, every geodesic that does not pass through the apex intersects itself exactly once.
For a smooth manifold, just round off the top.