User niek de kleijn - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T13:33:36Z http://mathoverflow.net/feeds/user/27224 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109570/can-the-level-set-of-a-critical-value-be-a-regular-submanifold Can the level set of a critical value be a regular submanifold? Niek de Kleijn 2012-10-13T23:56:00Z 2012-10-14T14:14:46Z <p>I know that there is a theorem that a non-empty level set of a regular value of a smooth function $f:M\rightarrow\mathbb{R}$ on a smooth manifold is a regular submanifold (or embedded submanifold) of codimension 1.</p> <p>Now I wonder if there is also a condition in which the converse holds true. I mean suppose $c\in\mathbb{R}$ is a critical value of $f$ is there a condition under which you know the level set of $c$ is not a regular submanifold. </p> <p>If $g:M\rightarrow\mathbb{R}$ is defined by mapping all of $M$ to $0$ then it seems to me that $0$ is a critical value of $g$. Now $g^{-1}(0)=M$ so definitely a regular submanifold of $M$. So I know that it is at least not true without an extra condition. </p> <p>Also if the function is second degree polynomial from $\mathbb{R}$ to $\mathbb{R}$ then the level set of a critical value is just a point which would again be a regular submanifold. </p> <p>If there is no such general condition on the function $f$ or the critical value then how in general does one go about showing that a critical level set is or is not a regular submanifold?</p> <p>Thanks in advance </p> http://mathoverflow.net/questions/109570/can-the-level-set-of-a-critical-value-be-a-regular-submanifold Comment by Niek de Kleijn Niek de Kleijn 2012-10-14T14:23:53Z 2012-10-14T14:23:53Z @Andrej I found this theorem in &quot;An Introduction to Manifolds&quot; by Loring W. Tu. In the capter on submanifolds. The constant rank is indeed a generalization of this theorem. http://mathoverflow.net/questions/109570/can-the-level-set-of-a-critical-value-be-a-regular-submanifold/109571#109571 Comment by Niek de Kleijn Niek de Kleijn 2012-10-14T14:22:18Z 2012-10-14T14:22:18Z Thank you, I didn't think to check the book by Arnold, Gussein-Zade and Varchenko even though I had it lying around the house!