Uniqueness of distance realizing geodesic in hyperbolic surface. - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-21T19:51:48Z http://mathoverflow.net/feeds/question/95724 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95724/uniqueness-of-distance-realizing-geodesic-in-hyperbolic-surface Uniqueness of distance realizing geodesic in hyperbolic surface. Bidyut Sanki 2012-05-02T05:22:59Z 2012-05-02T06:47:32Z <blockquote> <p><strong>Possible Duplicate:</strong><br> <a href="http://mathoverflow.net/questions/95640/hyperbolic-surfaces" rel="nofollow">Hyperbolic surfaces</a> </p> </blockquote> <p>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. </p> http://mathoverflow.net/questions/95724/uniqueness-of-distance-realizing-geodesic-in-hyperbolic-surface/95728#95728 Answer by Sam Nead for Uniqueness of distance realizing geodesic in hyperbolic surface. Sam Nead 2012-05-02T06:40:45Z 2012-05-02T06:47:32Z <p>For the pants, yes. In general, no. To prove this for the pants, classify <em>all</em> 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. </p> <p>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 <a href="http://mathoverflow.net/faq#whatnot" rel="nofollow">http://mathoverflow.net/faq#whatnot</a></p>