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.
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.
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.