applications of the sphere theorem I am looking for interesting applications of the 1/4-pinched 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.
 A: An application occurs in the study of asymptotic behavior of complete manifolds with certain curvature decay.
Let M be a n-dimensional complete non-compact manifold.
Suppose that


*

*M is simply-connected at infinity, 

*the sectional curvatures of M go to zero at infinity,

*there exists a foliation of (n-1)-dimensional sub-manifolds on the ends of M 

*these sub-manifolds have controlled second fundamental form,


then you may use Gauss equation and the differential sphere theorem to say that these sub-manifolds are diffeomorphic to the sphere.
A: 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 by Abresch and Meyer. 
More recently Brendle-Schoen used Ricci flow to prove a definitive differentible sphere theorem; this and closely related work by Bohm-Wilking are (in my view) the most spectacular applications of Ricci flow beyond dimension three.
