Set of spheres

This article deals with the theorem of spheres from Riemann's differential geometry. For the set of spheres from the topology, see under set of spheres (topology) .

The theorem of spheres is an important result of global Riemannian geometry . After preliminary work by Harry Rauch , Wilhelm Klingenberg and Marcel Berger proved this proposition in 1961.

Set of spheres

(Classical) set of spheres

Let be an n-dimensional, compact , simply connected , Riemannian manifold for its section curvature ${\displaystyle (M,g)}$ ${\displaystyle K}$

${\displaystyle 0<h<K\leq 1}$

with applies. Then it is homeomorphic to the sphere . ${\displaystyle \textstyle h={\frac {1}{4}}}$ ${\displaystyle M}$

Differentiable set of spheres

If the Riemannian manifold or its sectional curvature fulfills the same requirements as in the (classical) theorem of spheres, then it is diffeomorphic to the sphere, which is equipped with the normal differentiable structure . ${\displaystyle (M,g)}$ ${\displaystyle M}$

Origin of the sentence

The set of spheres was proven by Harry Rauch in 1951 for . Wilhelm Klingenberg linked this problem with the location of the cut . In the case that the manifold has straight dimensions and satisfies the above inequality with regard to the section curvature, the distance to the point of intersection was greater than the same (Lemma von Klingenberg). With this statement, Klingenberg proved the theorem of spheres for and even dimension. With the help of Toponogov's theorem and Klingenberg's lemma just mentioned, in 1960 Marcel Berger proved the theorem of spheres for and even dimension. In 1961 Klingenberg was able to prove the mentioned lemma for odd dimensions. The proof for odd dimensions is much more complicated and uses Morse theory . This completed the proof of the theorem of spheres. Tsukamoto was able to show that Toponogov's theorem is not necessary for the proof of the theorem of spheres. ${\displaystyle \textstyle h\sim {\frac {3}{4}}}$ ${\displaystyle \pi }$ ${\displaystyle h\,\sim \,0.55}$ ${\displaystyle \textstyle h={\frac {1}{4}}}$

In 2007 Simon Brendle and Richard Schoen succeeded in proving that under the above conditions the manifold is even diffeomorphic to the sphere. ${\displaystyle M}$

Auxiliary statements

In this section a few more statements are shown which are important for the proof of the theorem of spheres. Klingenberg's first lemma given here corresponds to the one from the section above.

Lemma from Klingenberg

Let be a compact, simply connected, Riemannian manifold, for whose intersection curvature the inequality ${\displaystyle M}$ ${\displaystyle K}$

${\displaystyle {\frac {1}{4}}<K\leq 1}$

applies. Then follows

${\displaystyle i(M)\geq \pi ,}$

where means the shortest distance to a next cut location. This is also called the injective radius of ${\displaystyle i(M)}$ ${\displaystyle M.}$

Existence of hemispheres

Let be an n-dimensional, compact, simply connected, Riemannian manifold, for the curvature of which holds, and be such that holds. Then follows ${\displaystyle M}$ ${\displaystyle \textstyle {\frac {1}{4}}<\delta \leq K\leq 1}$ ${\displaystyle p,q\in M}$ ${\displaystyle d(p,q)=\operatorname {diam} (M)}$

${\displaystyle M=B_{\rho }(p)\cup B_{\rho }(q),}$

where the open geodesic ball is denoted by radius and center . The function gives the diameter of the Riemann manifold. ${\displaystyle B_{\rho }(p)\subset M}$ ${\displaystyle {\tfrac {\pi }{2}}{\sqrt {\delta }}<\rho <\pi }$ ${\displaystyle p\in M}$ ${\displaystyle \operatorname {diam} }$

Existence of an equator

Under the assumptions made for the existence of hemispheres, there is a unique point for each geodetic with the length and with the starting point , so that ${\displaystyle \rho }$ ${\displaystyle p}$ ${\displaystyle k}$

${\displaystyle d(p,k)=d(q,k)}$

applies. Likewise, for every geodetic with starting point and length , there is a unique point that is equidistant from and . The function is the distance function which is induced by the Riemannian metric. ${\displaystyle q}$ ${\displaystyle \rho }$ ${\displaystyle l}$ ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle d(.,.)}$

Further remarks

Constructed homeomorphism

In the proof of the theorem of spheres, Berger constructed a function of which he showed that it is a homeomorphism. Let be an isometry for one and be the antipodal point of . The function is now defined by ${\displaystyle h\colon \mathbb {S} ^{n}\to M}$ ${\displaystyle I\colon T_{\bar {p}}\mathbb {S} ^{n}\to T_{p}M}$ ${\displaystyle {\bar {p}}\in \mathbb {S} ^{n}}$ ${\displaystyle {\bar {q}}=-{\bar {p}}}$ ${\displaystyle {\bar {p}}}$ ${\displaystyle h\colon \mathbb {S} ^{n}\to M}$

${\displaystyle x\mapsto {\begin{cases}p,&x={\bar {p}}\in \mathbb {S} ^{n}\\\exp _{p}\left({\frac {d(x,{\bar {p}})}{\pi /2}}\cdot (f\circ I\circ \exp _{\bar {p}}^{-1})(x)\right),&d(x,{\bar {p}})\leq {\tfrac {\pi }{2}}\\\exp _{q}\left({\frac {d(x,{\bar {q}})}{\pi /2}}\cdot (\exp _{q}^{-1}\circ \exp _{p}\circ f\circ I\circ \exp _{\bar {p}}^{-1})(x)\right),&d(x,{\bar {q}})\leq {\tfrac {\pi }{2}}\\q,&x={\bar {q}}\in \mathbb {S} ^{n}.\end{cases}}}$

The function is the exponential mapping and is the distance function which is induced by the Riemannian metric. ${\displaystyle \exp }$ ${\displaystyle d(.,.)}$

Optimal limit

The complex projective space for is compact and simply connected and the curvature of section satisfies the inequality . However, it is known that complex projective space is not homeomorphic to the sphere. That means, with an even dimension is the optimal bound. If the dimension is odd, it is known that the theorem also applies to. However, the optimal bound has not yet been found. The sentence is even correct for Dimension . ${\displaystyle \mathbb {C} \mathbb {P} ^{n}}$ ${\displaystyle n>1}$ ${\displaystyle \textstyle {\frac {1}{4}}\leq K\leq 1}$ ${\displaystyle n\geq 4}$ ${\displaystyle {\tfrac {1}{4}}}$ ${\displaystyle \textstyle {\frac {1}{4}}\leq K\leq 1}$ ${\displaystyle n=2,3}$ ${\displaystyle \textstyle 0\leq K\leq 1}$

Hamilton theorem

Twenty-five years before the differentiable theorem of spheres could be proven, Richard S. Hamilton published a theorem in 1982, which he derived from the (topological) theorem of spheres using techniques from the theory of partial differential equations. The statement of the sentence reads:

Let be a compact, simply connected, Riemannian manifold of dimension three with strictly positive Ricci curvature. Then is diffeomorphic to the sphere . ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle \mathbb {S} ^{3}}$

literature

• Manfredo Perdigão do Carmo: Riemannian Geometry , Birkhäuser, Boston 1992, ISBN 0-8176-3490-8
• Simon Brendle The set of spheres in Riemannian geometry , Annual Report DMV, Volume 113, 2011, Issue 3, pp. 123-138

Individual evidence

1. ^ Rauch, HE, A contribution to differential geometry in the large, Ann. of Math. 54: 38-55 (1951)
2. ^ Klingenberg, W., Contributions to riemannian Geometry in the large, Ann. of Math. 69: 654-666 (1959).
3. ↑ Berger, M., Les variétés Riemannienes (1/4) -pincées, Ann. Scuola Norm. Sup. Pisa, Ser. III, 14, 161-170 (1960)
4. ↑ Klingenberg, W., On Riemannian Manifolds with Positive Curvature, Comm. Math. Helv. 35 (1961), 47-54.
5. ↑ Brendle, Schoen, Manifolds with 1-4 pinched curvature are space forms , Journal of the AMS, Vol. 22, 2009, p. 287, Classification of manifolds with 1-4 pinched curvature , Acta Mathematica, Vol. 200, 2008, P. 1
6. ^ Richard S. Hamilton: Three manifolds with positive Ricci curvature. In: Journal of Differential Geometry. 17, No. 2, 1982, pp. 255-306.