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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3$\begingroup$ No. The closed geodesic in a homotopy class is unique. $\endgroup$– Lee MosherCommented Sep 11, 2018 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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$\begingroup$ Here's a link to A primer on mapping class groups by Farb and Margolit: maths.ed.ac.uk/~v1ranick/papers/farbmarg.pdf $\endgroup$– NealCommented Sep 11, 2018 at 13:22