# Example of a manifold with not “closed” geodesic

From my very limited experience and understanding, I have come to realize that some people study "closed" geodesics. I understand that to mean that for some a, b x(a)=x(b) for the geodesic curve x. In what situations would a manifold admit a geodesic that is not closed? Perhaps I am just not "seeing" it :-). Thank you for any help.

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How about $\mathbb R^n$? Then the geodesics are straight lines. –  Jeff Nov 29 '11 at 20:58
Yes, thank you. –  Aspar T. Ame Nov 29 '11 at 21:24

There is an additional condition that belongs here. A geodesic is called closed when your $x(a) = x(b)$ AND $x'(a) = x'(b),$ meaning that it makes a closed smooth curve, and keeps going over the same same set of points forever.

It is easy to find manifolds with self-intersecting geodesics that are not closed geodesics.

The traditional example, the surface of revolution $z = x^2 + y^2,$ is an exercise in do Carmo. If a geodesic goes through $(0,0,0)$ it just goes on forever. Otherwise, it intersects itself infinitely often, but is not closed. Here we go, pages 258-260 in Differential Geometry of Curves and Surfaces

I met do Carmo at a conference once. He sent me Celso Costa's dissertation. Which is why there are bound copies in the libraries of U.C. Berkeley and U.C. San Diego.

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Thank you; very good point about x'(a) = x'(b) –  Aspar T. Ame Dec 12 '11 at 3:47