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?
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.