most recent 30 from http://mathoverflow.net 2013-05-24T05:58:26Z http://mathoverflow.net/feeds/question/52554 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/52554/hypersurfaces-and-elliptic-points T-' 2011-01-19T22:29:26Z 2011-01-20T03:33:08Z

<p>I'm reading a paper, in which we have $M^n$ an n-dimensional compact hypersurface embedded in $\mathbb{R}^{n+1}$. We take the scalar cuvature $R$ to be the elementary symmetric polynomial of degree 2 in the principal curvatures of $M$. We know that $R$ is constant.</p> <p>The author then says "As $M$ has one elliptic point, $R$ is a positive constant and the mean curvature is positive somewhere".</p> <p>I'm lost here - Why does $M$ have an elliptic point? And how does this affect $R$ and the mean curvature?</p> <p>Thanks for any help.</p>

http://mathoverflow.net/questions/52554/hypersurfaces-and-elliptic-points/52573#52573 Igor Rivin 2011-01-20T03:33:08Z 2011-01-20T03:33:08Z

<p>Elliptic point is, by definition, a point where all the principal curvatures are positive, hence $R$ is positive. A point of maximal distance from some far-away basepoint is elliptic.</p>