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.
[Math] applications of the sphere theorem
dg.differential-geometry
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The new proofs add a layer of complication that is unnecessary for your question. From Cheeger and Ebin, Comparison Theorems in Riemannian Geometry first paragraph of Chapter 6:
The symmetric spaces of positive curvature are known to admit metrics such that $1 \geq K_M \geq > \frac{1}{4}$; see Example 3.38. In fact, we will prove that any riemannian manifold with $1 \geq K_M > \geq \frac{1}{4},$ which is not a sphere, is isometric to one of these spaces.
Back in chapter 3, page 73,
A complete classification of symmetric spaces is available (Helgason [1962]). In particular, the only simply connected symmetric spaces having positive curvature are the spheres of constant curvatures, complex and quaternionic projective spaces, and the Cayley plane. These are sometimes referred to as the rank one symmetric spaces, and except for the spheres they have canonical metrics varying between $\frac{1}{4}$ and $1.$ As an example, we will compute the curvature of complex projective space. The calculations for the other rank one spaces are similar.
Then they begin the example, which is numbered Example 3.38. The punchline is that $\alpha$ and $\beta$ are orthogonal unit vectors, $J(\alpha)$ and $\beta$ are unit real vectors, so their real inner product lies between $-1$ and $1$ by Cauchy-Schwarz, followed by an identity about the sectional curvature that I shall write as $$ K_{\alpha \beta} = \frac{1}{4} + \frac{3}{4} \langle J(\alpha), \beta \rangle^2. $$
Chern-Gauß-Bonnet implies that the volume of a hyperbolic manifold is a constant multiple of its Euler characteristics, with the constant factor depending on dimension only.
In particular, a hyperbolic manifold with $$\mid\chi(M)\mid=1$$ necessarily is the hyperbolic manifold of minimal volume in its dimension. Ratcliffe, Tschantz and Everitt have used this to find the hyperbolic manifolds of minimal volume in dimensions 4 and 6.
Best Answer
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.