TITLE: applications of the sphere theorem
QUESTION [12 upvotes]: 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.

REPLY [5 votes]: 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.