User dmitri gekhtman - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T06:19:40Z http://mathoverflow.net/feeds/user/23743 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/28421/the-jacobi-identity-for-the-poisson-bracket/109831#109831 Answer by Dmitri Gekhtman for The Jacobi Identity for the Poisson Bracket Dmitri Gekhtman 2012-10-16T16:28:57Z 2012-10-16T16:28:57Z <p>Here's the way it's done in John Lee's Smooth Manifolds book:</p> <p>$${\small \iota_{X_{\lbrace f,g\rbrace}}\omega = d\lbrace f,g \rbrace=d(X_gf) = d(\mathcal{L}_{X_g}f) =\mathcal{L}_{X_g}df=\mathcal{L}_{X_g}(\iota_{X_f}\omega)=\mathcal{\iota}_{[X_g,X_f]}\omega + \mathcal{L}_{X_g}\omega=\mathcal{\iota}_{[X_g,X_f]}\omega}$$ which by nondegeneracy of $\omega$ implies the desired result.</p> http://mathoverflow.net/questions/103147/a-strange-question-about-closed-geodesics-on-a-closed-manifold A strange question about closed geodesics on a closed manifold Dmitri Gekhtman 2012-07-26T02:26:40Z 2012-07-26T10:29:53Z <p>I'm studying a particular kind of curve evolution on Riemannian manifolds. It would help me to know the answer to the following kinda weird question:</p> <p>Does there exist a closed Riemannian manifold $M$ and a pair of distinct closed geodesics $\gamma, \alpha$ in $M$ satisfying the follow properties?</p> <p>(1) $\gamma$ and $\alpha$ are of the same length. (Call the length $l$.)</p> <p>(2) $\gamma$ is isolated in the space of loops of length $l$.</p> <p>(3) For any $\varepsilon>0$, there exists a path of loops, each of length between $l$ and $l+\varepsilon$, starting at $\gamma$ and ending at $\alpha$.</p> <p>I suspect (and hope) that the answer is no.</p> <p>Details: The loop space I'm working with is $C^0(S^1,M)$ with the compact-open topology. In (2), I mean that $\gamma$ has a neighborhood in the loop space in which the only loops of length $l$ are reparameterizations of $\gamma$. The path in (3) of course has to be a path of rectifiable loops. </p> <p>Thanks,</p> <p>Dmitri </p> http://mathoverflow.net/questions/97083/are-isospectral-manifolds-necessarily-homeomorphic Are isospectral manifolds necessarily homeomorphic? Dmitri Gekhtman 2012-05-16T03:58:05Z 2012-05-16T04:30:30Z <p>It's known that there are pairs of closed Riemannian manifolds which are isospectral but not isometric. Is it known if there are closed Riemannian manifolds which are isospectral but not homeomorphic? (By isospectral, I mean that the Laplace-Beltrami operator on functions has the same spectrum on both manifolds.)</p> <p>Thanks,</p> <p>Dmitri</p> http://mathoverflow.net/questions/28421/the-jacobi-identity-for-the-poisson-bracket/109831#109831 Comment by Dmitri Gekhtman Dmitri Gekhtman 2012-10-18T02:39:40Z 2012-10-18T02:39:40Z Oh woops. Didn't get to the punchline. I showed $$X_{\lbrace f,g\rbrace} = [X_g,X_f].$$ From here, do what Zack said. http://mathoverflow.net/questions/103147/a-strange-question-about-closed-geodesics-on-a-closed-manifold/103180#103180 Comment by Dmitri Gekhtman Dmitri Gekhtman 2012-07-26T15:02:37Z 2012-07-26T15:02:37Z @Anton Petrunin Thanks for your answer! It's not a homework problem, but you're right that I should have thought about the question a bit harder before asking on MO. I guess it's not as difficult as I expected. I don't know why your answer was downvoted.