I am looking for interesting applications of the [1/4-pinched sphere theorem](http://www.en.wikipedia.org/wiki/Sphere_theorem). The theorem says: A compact, simply connected riemannian manifold whose sectional curvature K satisfies $1/4 < K \leq$ 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.



applications of the sphere theorem
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. 


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 [the survey](http://library.msri.org/books/Book30/files/abresch.pdf) by Abresch and Meyer. 


More recently Brendle-Schoen used Ricci flow to prove a definitive [differentible sphere theorem](http://en.wikipedia.org/wiki/Sphere_theorem); this and closely related [work](http://annals.math.princeton.edu/2008/167-3/p10) by Bohm-Wilking are (in my view) the most spectacular applications of Ricci flow beyond dimension three.


