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> **Possible Duplicate:**  
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> [Hyperbolic surfaces](https://mathoverflow.net/questions/95640/hyperbolic-surfaces) 
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Given a hyperbolic surface S with geodesic boundary. Let a and b be two distinct simple closed geodesic boundaries. Does there exist a unique distance realizing geodesic in S?
 (1) for S is a pair of pants. 
(2) S is any hyperbolic surface with boundary. 



Uniqueness of distance realizing geodesic in hyperbolic surface.
For the pants, yes. In general, no. To prove this for the pants, classify *all* geodesic arcs and just observe the result. There are many ways to find a "no" example in the general case; the first one that came to my mind was taking a double cover. 


EDIT - I see that this is a near-duplicate of a closed question. You could improve your question by giving some motivation. Reading the FAQ will be very useful in writing questions that get good answers. In particular please see <https://mathoverflow.net/faq#whatnot>


