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Let $a$ and $b$ be non-intersecting closed geodesics on a hyperbolic surface. Can these curves be homotopied to transversely intersect but still be geodesics?

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    $\begingroup$ No. The closed geodesic in a homotopy class is unique. $\endgroup$ – Lee Mosher Sep 11 '18 at 12:14
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No. The "geodesic parametrization" of a curve is unique (up to pre-composition with a rotation) in the curve's homotopy class. You can find more information in the book "A primer on mapping class groups".

Edit: and Lee beat me to the answer by just two minutes! I'll leave this here for the reference.

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