Hypersurfaces and Elliptic Points - MathOverflow 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 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 Answer by Igor Rivin for Hypersurfaces and Elliptic Points 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>