most recent 30 from http://mathoverflow.net 2013-05-21T14:27:35Z http://mathoverflow.net/feeds/question/99296 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/99296/embedding-torus geomtfree 2012-06-11T13:16:54Z 2012-08-17T14:27:17Z

<p>hi everybody could anyone please help me? why is it impossible to embed a torus in R^3 with index 1 ( usual euclidean space with index 1 as a semi-riemannian manifold) as a semi-riemannian submanifold?</p>

http://mathoverflow.net/questions/99296/embedding-torus/99306#99306 Lee Mosher 2012-06-11T16:08:04Z 2012-06-11T16:08:04Z

<p>I suppose that by "usual" you mean $\mathbb{R}^3$ with the semi-Riemannian metric $dx^2 + dy^2 - dz^2$?</p> <p>If so, for any compact surface $S$ embedded in $\mathbb{R}^3$, since the metric is invariant under translation on $\mathbb{R}^3$ you are allowed to translate $S$ so that it is tangent to the light cone $x^2 + y^2 - z^2 = 0$. At any point of tangency the restricted metric is indefinite. To do this translation, first translate $S$ so that it is strictly inside the light cone, then let $S$ gently descend until the first moment that it touches the light cone.</p>

http://mathoverflow.net/questions/99296/embedding-torus/99339#99339 geomtfree 2012-06-12T07:05:05Z 2012-06-12T07:05:05Z

<p>thanx. but i didn't understand. let me ask another question. why with this metric R^3 doesn't have any compact semi-riemannian submanifold?</p>