Can the geodesic flow be preserved by an inhomogeneous rescaling of a cross section? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T08:05:08Z http://mathoverflow.net/feeds/question/26951 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/26951/can-the-geodesic-flow-be-preserved-by-an-inhomogeneous-rescaling-of-a-cross-secti Can the geodesic flow be preserved by an inhomogeneous rescaling of a cross section? Steve Huntsman 2010-06-03T19:17:20Z 2010-06-17T21:22:15Z <p>Let $M$ be a compact Riemannian manifold with metric $g$ and associated Riemannian volume $\nu$ and <a href="http://en.wikipedia.org/wiki/Geodesic#Geodesic_flow" rel="nofollow">geodesic flow</a> $G_t : UTM \rightarrow UTM$, where the <a href="http://en.wikipedia.org/wiki/Unit_tangent_bundle" rel="nofollow">unit tangent bundle</a> is indicated. Let $X_j \subset UTM$ for $1 \le j \le n$ be open disjoint codimension one submanifolds transversal to $G_t$, i.e., local cross sections (<a href="http://books.google.com/books?id=1K66IrjWbgwC&amp;lpg=PP1&amp;ots=EPHHQe4_Ip&amp;dq=Geodesic%2520%25EF%25AC%2582ows&amp;pg=PA49#v=onepage&amp;q=lemma%25202.52&amp;f=false" rel="nofollow">a global cross section does not exist</a>). </p> <blockquote> <p>Is it possible to choose a metric $g'$ on $M$ with geodesic flow $G'_t = G_t$ and $\nu'_1(X_j) \equiv 1$?</p> </blockquote> <p>NB. Here $\nu'_1$ denotes the induced codimension one [relative] measure on $UTM$.</p> <p>This question was prompted by a helpful comment to <a href="http://mathoverflow.net/questions/26834/" rel="nofollow">this one</a>.</p> http://mathoverflow.net/questions/26951/can-the-geodesic-flow-be-preserved-by-an-inhomogeneous-rescaling-of-a-cross-secti/26969#26969 Answer by Steve Huntsman for Can the geodesic flow be preserved by an inhomogeneous rescaling of a cross section? Steve Huntsman 2010-06-03T20:58:22Z 2010-06-03T20:58:22Z <p>Unless I misunderstand this reference<sup>*</sup> after a cursory look, it appears that the answer is generally no: <a href="http://books.google.com/books?id=njIG5AqcQ2YC&amp;pg=PA217" rel="nofollow">"geodesic conjugacy" often implies isometry.</a> </p> <hr> <p><sup>*</sup> and anways, why is it that references seem so much easier to find once I've posted a question on MO?</p>