most recent 30 from http://mathoverflow.net 2013-05-25T07:09:51Z http://mathoverflow.net/feeds/user/8208 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/63345/famous-2d-riemannian-manifolds-with-non-constant-curvature/63457#63457 Giovanni Rastelli 2011-04-29T18:42:39Z 2011-04-29T18:42:39Z

<p>This is not an answer to the question but could be interesting as well, regarding non-constant curvature manifolds and Killing fields. If you take the tensor product of two Killing vectors, you obtain a second-degree (contravariant) Killing tensor. However, not all the second-degree Killing tensors can be obtained in this way, depending on the (pseudo-)Riemannian manifold. For example, in all constant curvature (pseudo-)Riemannian manifolds all the second-degree Killing tensors are linear combinations of symmetric tensor products of Killing vectors (and similarly for higher degree Killing tensors), Killing tensors of this type are called reducible. This is not true for manifolds with non constant curvature. The (very classical) example is the skew ellipsoid, which obviously does not admit Killing vectors but admits a second-degree Killing tensor. The existence of this tensor is connected with the integrability of the geodesic flow on the skew ellipsoid and the celebrated Jacobi's ellipsoidal coordinates. A kind of connection that holds for any (pseudo-)Riemannian manifold under certain conditions. As far I know, there is no rule to determine if a manifold of non-constant curvature admits or not non-reducible Killing tensors of some degree.</p>

http://mathoverflow.net/questions/52830/curvature-and-riemannian-metric/62894#62894 Giovanni Rastelli 2011-04-25T06:34:44Z 2011-04-25T09:01:53Z

<p>This is maybe late for your seminar, but a classical textbook about Riemannian geometry, including relations between curvature and metric tensor, with any signature, is </p> <p>Riemannian geometry by L.P.Eisenhart</p> <p>Another very interesting book for you could be</p> <p>Spaces of constant curvature by J. A. Wolf</p> <p>Indeed, it seems that you are searching for the Riemannian manifolds whose metric element can be written as sum and/or difference of squares of coordinate differentials. This implies that the curvature is constant and equal to 0. As shown in Wolfs'book, this can be locally realized by several manifolds with different "global geometry". As example in dimension 2, $ds^2=dx^2+dy^2$ can be realized on the (Euclidean) plane, on the cylinder, on the torus, on the Moebius strip and on the Klein bottle, while $ds^2=dx^2-dy^2$ on the (Minkowski) plane, on the torus and on the Klein bottle.</p>

http://mathoverflow.net/questions/6379/what-is-an-integrable-system/34736#34736 Giovanni Rastelli 2010-08-06T06:47:35Z 2010-08-06T06:47:35Z

<p>After reading several books and articles about integrable systems, and after several years of work in the field, I consider particularly meaningful the following quotation from Frederic Helein's book <em>'Constant mean curvature surfaces, harmonic maps and integrable systems', Lectures in Mathematics, ETH Zurich, Birkhauser Basel (2001):</em></p> <p>"...working on completely integrable systems is based on a contemplation of some very exceptional equations which hide a Platonic structure: although these equations do not look trivial a priori, we shall discover that they are elementary, once we understand how they are encoded in the language of symplectic geometry, Lie groups and algebraic geometry. It will turn out that this contemplation is fruitful and lead to many results"</p>

http://mathoverflow.net/questions/63345/famous-2d-riemannian-manifolds-with-non-constant-curvature/63457#63457 Giovanni Rastelli 2011-05-03T18:53:25Z 2011-05-03T18:53:25Z

Dear Vladimir, it seems that this algorithm connects the dimension of the space of Killing tensors of any degree with their reducibility or not, at least locally. That procedure seems very much intriguing. Could you give precise references?