applications of the sphere theorem - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T22:24:45Z http://mathoverflow.net/feeds/question/60341 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/60341/applications-of-the-sphere-theorem applications of the sphere theorem whatever 2011-04-02T03:00:58Z 2011-04-18T12:36:35Z <p>I am looking for interesting applications of the <a href="http://www.en.wikipedia.org/wiki/Sphere_theorem" rel="nofollow">1/4-pinched sphere theorem</a>. The theorem says: A compact, simply connected riemannian manifold whose sectional curvature K satisfies $1/4 &lt; K \leq$ 1 (possibly after multiplying the metric by a constant) is homeomorphic (recently extended to "diffeomorphic") to the sphere. I just wanted to know: is it just a beautiful theorem or can you use it in concrete situations to derive some conclusions difficult to see otherwise? I am interested in this just because I am curious, I do not have any specific purpose in mind.</p> http://mathoverflow.net/questions/60341/applications-of-the-sphere-theorem/60438#60438 Answer by Igor Belegradek for applications of the sphere theorem Igor Belegradek 2011-04-03T14:17:24Z 2011-04-03T14:17:24Z <p>The main theme of global Riemannian geometry is to derive topological conclusions from geometric assumptions. Sphere theorems provide various assumptions under which a manifold is (homeomorphic, diffeomorphic, or almost isometric) to a sphere. </p> <p>The significance of sphere theorems is not in their applications or implications but in the beautiful mathematics they generated. Tools developed to prove various sphere theorems is a backbone of modern comparison geometry, and a great place to learn about it is <a href="http://library.msri.org/books/Book30/files/abresch.pdf" rel="nofollow">the survey</a> by Abresch and Meyer. </p> <p>More recently Brendle-Schoen used Ricci flow to prove a definitive <a href="http://en.wikipedia.org/wiki/Sphere_theorem" rel="nofollow">differentible sphere theorem</a>; this and closely related <a href="http://annals.math.princeton.edu/2008/167-3/p10" rel="nofollow">work</a> by Bohm-Wilking are (in my view) the most spectacular applications of Ricci flow beyond dimension three.</p> http://mathoverflow.net/questions/60341/applications-of-the-sphere-theorem/62115#62115 Answer by Chih-Wei Chen for applications of the sphere theorem Chih-Wei Chen 2011-04-18T12:36:35Z 2011-04-18T12:36:35Z <p>An application occurs in the study of asymptotic behavior of complete manifolds with certain curvature decay.</p> <p>Let M be a n-dimensional complete non-compact manifold. Suppose that</p> <ol> <li>M is simply-connected at infinity, </li> <li>the sectional curvatures of M go to zero at infinity,</li> <li>there exists a foliation of (n-1)-dimensional sub-manifolds on the ends of M </li> <li>these sub-manifolds have controlled second fundamental form,</li> </ol> <p>then you may use Gauss equation and the differential sphere theorem to say that these sub-manifolds are diffeomorphic to the sphere.</p>