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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.

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    $\begingroup$ Well, it is a beautiful and natural theorem... Pleasure should be counted as an application! $\endgroup$ – Mariano Suárez-Álvarez Apr 2 '11 at 3:57
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    $\begingroup$ One the top of this web page, there is a link labeled "how to ask". Please read the page that is linked there, and revise your question. $\endgroup$ – S. Carnahan Apr 2 '11 at 4:17
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    $\begingroup$ When you have finished, please flag for moderator attention, so the question can be reopened. $\endgroup$ – S. Carnahan Apr 2 '11 at 4:18
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    $\begingroup$ I personally think that this is not such a bad question that it needed to be insta-closed. I'd be interested in hearing answers! $\endgroup$ – Andy Putman Apr 2 '11 at 14:59
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    $\begingroup$ @whatever, read Comparison Theorems in Riemannian Geometry, by Jeff Cheeger and David Ebin, undergraduate/beginning graduate level. There are more recent books with similar material as well. This will satisfy any curiosity you might have about the place of the two Sphere Theorems, homeomorphic and diffeomorphic, as to difficulty and place in mathematics. Chapter 7 discusses alternate differentiable structures. Note that Calabi and Gromoll proved the Differentiable version in 1966, but needed a pinchng constant depending on dimension.. Finally, actually read Brendle and Schoen. $\endgroup$ – Will Jagy Apr 2 '11 at 19:54
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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.

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    $\begingroup$ Very nice answer. $\endgroup$ – Deane Yang Apr 3 '11 at 14:20
  • $\begingroup$ Thanks guys, those were really very helpful. I am particularly reading the survey by Abresch and Meyer now. I saw someone comment on the meta thread about the application of the sphere theorem to the aphericity of knot complements....can anyone tell me a reference for this? $\endgroup$ – whatever Apr 7 '11 at 0:39
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    $\begingroup$ @whatever: I doubt asphericity of knowt complements is related to this sphere theorem. It might be related to another one: en.wikipedia.org/wiki/Sphere_theorem_%283-manifolds%29 $\endgroup$ – Igor Belegradek Apr 7 '11 at 3:46
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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

  1. M is simply-connected at infinity,
  2. the sectional curvatures of M go to zero at infinity,
  3. there exists a foliation of (n-1)-dimensional sub-manifolds on the ends of M
  4. 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.

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