Sphere theorem

The spheres theorem is an important result from the global Riemannian geometry. After preliminary work by Harry smoke Wilhelm Klingenberg and Marcel Berger proved this theorem in 1961.

• Lemma 3.1 of Klingenberg
• 3.2 existence of hemispheres
• 3.3 existence of an equator

• 4.1 constructed homeomorphism

Spheres set

( Classic ) spheres set

Be an n-dimensional, compact, simply connected Riemannian manifold whose sectional curvature for

With applies. Then is homeomorphic to the sphere.

Differentiable sphere theorem

Meets the Riemannian manifold whose sectional curvature or the same conditions as in (classical) spheres set, it is diffeomorphic to the sphere, which is equipped with the standard differentiable structure.

Emergence of the sentence

The spheres theorem was proved by Harry Rauch in 1951 for. Wilhelm Klingenberg brought this issue to the slice location in context. In the case that the manifold has dimension and just above inequality with respect to an average curvature, was the distance to the intersection location greater than or equal ( Lemma Klingenberg ). With this statement Klingenberg proved the spheres set for straight and dimension. With the help of the set of Toponogov and the just-mentioned lemma of Klingenberg proved 1960 Marcel Berger spheres set for and even dimension. In 1961, Klingenberg could prove the lemma mentioned also for odd dimension. The proof for odd dimensions is much more complicated and uses Morse theory. This completed the proof of the sphere theorem. Tsukamoto was able to show that the set of Toponogov for the proof of the sphere theorem is not necessary.

In 2007, succeeded Simon Brendle and Richard Schoen to prove that under the above conditions, the manifold is even diffeomorphic to the sphere.

Auxiliary statements

In this section, some statements still be shown, which are important for the proof of the sphere theorem. The specified here as the first lemma of Klingenberg corresponds to that in the above section.

Lemma of Klingenberg

Be a compact, simply connected Riemannian manifold whose sectional curvature for the inequality

Applies. Then follows

With the shortest distance to the next intersection location says. This is also called the injective radius of

Existence of hemispheres

Be an n-dimensional, compact, simply connected Riemannian manifold whose sectional curvature is valid for, and be so true. Then follows

Where the open geodesic ball designated radius and with center point. The function gives the diameter of the Riemannian manifold.

Existence of an equator

Under the assumptions made ​​about the existence of hemispheres exists for every geodesic of length and starting point of a unique point, so that

Applies. Just as is true for every geodesic with starting point and length that a unique point exists which is equidistant from and. The function of the distance function that is induced by the Riemannian metric.

Additional comments

Constructed homeomorphism

Berger constructed in the proof of the sphere theorem, a function of which he showed that it is a homeomorphism. Be a an isometry and be the antipodal point. The function is now defined by

The function is the exponential function and is the distance, which is induced by the Riemannian metric.

The complex projective space is compact and simply connected and the sectional curvature satisfies the inequality. However, it is known that the complex projective space is not homeomorphic to the sphere. That is, in even dimension is the optimal barrier. In odd dimension is known that the rate also applies. However, the optimal barrier has not yet been found. For dimension of the movement is in fact correct.

Set of Hamilton

Twenty-five years before the differentiable sphere theorem was proved Richard Hamilton published in 1982 a set, which he derived with the help of techniques from the theory of partial differential equations of the ( topological ) spheres set. The statement of the theorem is:

Be a compact, simply connected Riemannian manifold of dimension three with strictly positive Ricci curvature. Then is diffeomorphic to the sphere.