Zeta function for curves in a manifold - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T10:33:21Z http://mathoverflow.net/feeds/question/10403 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/10403/zeta-function-for-curves-in-a-manifold Zeta function for curves in a manifold Ilya Nikokoshev 2010-01-01T20:51:41Z 2010-01-05T20:57:38Z <h3>Motivation</h3> <p>In the analogy between prime numbers and knots, the prime number is thought sometimes as the circle of length <code>$l([p]) = \text{log}\,p$</code>. This is so you can express the zeta function as <code>$$\zeta(s) = \sum_{D\ge0} e^{-l(D)s}$$</code></p> <p>where the sum goes over effective divisors on <code>$\text{Spec}\,\mathbb Z$</code> and length is extended there by additivity. Similarly, you can do it to rewrite Dedekind zeta function for other number fields.</p> <h3>Question</h3> <p>I wonder, <strong>what is the right analogue of above formula</strong> for a manifold with metric? Perhaps:</p> <ol> <li>integration over all closed curves of the expression <code>$e^{-l(D)s}$</code></li> <li>summation over positive sums of classes of closed geodesics.</li> </ol> <p>I think I've heard something about definition 2, but I suspect if the two above are defined correctly they will be the same. Is it possible to formalize this definition? Do different formalizations lead to the same zeta-function?</p> <h3>Updates</h3> <p>Yes, I think this should be related to Laplacians, Selberg trace formula and dynamical system zetas. What I said I've heard about definition 2 was probably the <a href="http://en.wikipedia.org/wiki/Selberg%5Fzeta%5Ffunction" rel="nofollow">Selberg zeta</a>, but I can't say it clearly, hence questions.</p> http://mathoverflow.net/questions/10403/zeta-function-for-curves-in-a-manifold/10829#10829 Answer by Richard Montgomery for Zeta function for curves in a manifold Richard Montgomery 2010-01-05T18:17:08Z 2010-01-05T18:17:08Z <p>In a slightly different spirit, but still carrying the analogy between closed curves and primes, The Selberg trace formula relates a sum over the lengths of closed geodesics on a hyperbolic surface (compact or no) to the spectrum of the Laplacian on said surface. Googling `Selberg trace wiki' will get you started here.</p> <p>This formula has provided inspiration for Guillemin and collaborators working in spectral geometry and micro-local analysis. </p> http://mathoverflow.net/questions/10403/zeta-function-for-curves-in-a-manifold/10841#10841 Answer by Thomas Sauvaget for Zeta function for curves in a manifold Thomas Sauvaget 2010-01-05T20:38:56Z 2010-01-05T20:38:56Z <p>There's been a lot of work since Smale's idea of a dynamical zeta function for general flows (in particular geodesic flows). A good starting point would be this 12 year old <a href="http://www.dma.ens.fr/~baladi/etds.ps" rel="nofollow">review by Baladi</a>. There is a large and more recent literature but I'm no expert, although this <a href="http://www.mat.uniroma2.it/~liverani/Lavori/live0602.pdf" rel="nofollow">other review by Liverani and Tsuji</a> is probably not far from current knowledge.</p> <p>There's also a whole branch of physics around those ideas, indeed related to the spectrum of the Laplacian and applications to quantum physics and statistical physics. <a href="http://chaosbook.org/" rel="nofollow">This nice physics book</a> is a good start (in particular if you read the quote of Smale at page 3 of <a href="http://chaosbook.org/chapters/det.pdf" rel="nofollow">this chapter</a>, and then remark 19.2 at the very end of that chapter you'll get a quick sense of the stuff you've aked for).</p>