Cheeger's theorem of finiteness

Cheeger's finiteness theorem is a theorem from the mathematical sub-area of differential geometry , which makes a statement about the number (and especially the finiteness) of manifolds with given diameter, volume and curvature bounds.

Cheeger's Finiteness Theorem

For given positive numbers there is only a finite number of diffeomorphy types -dimensional Riemannian manifolds with ${\displaystyle n,D,V,K}$ ${\displaystyle C(n,D,V,K)}$ ${\displaystyle n}$ ${\displaystyle M}$

${\displaystyle \mathrm {diam} (M)\leq D,\mathrm {vol} (M)\geq V,-K\leq \mathrm {sec} (M)\leq K.}$

Here the volume , the diameter and the sectional curvatures denote the Riemann manifold . ${\displaystyle \operatorname {vol} (M)}$ ${\displaystyle \operatorname {diam} (M)}$ ${\displaystyle \sec(M)}$ ${\displaystyle M}$

history

The finiteness theorem first appeared in Cheeger's dissertation, at that time initially for finiteness of homeomorphism types and only under additional conditions also for diffeomorphism types. The finiteness theorem in its above form appears in Cheeger-Ebin and with proof as well as an explicit estimate for in Peters. Further proof as well as numerous generalizations and applications can be found in Chapter 8 of Gromov's book. ${\displaystyle C(n,D,V,K)}$

Remarks

Cheeger's proof relied largely on a lower bound on the radius of injectivity . He proved that under the assumptions a lower bound for the volume is equivalent to a lower bound for the injectivity radius. ${\displaystyle \operatorname {diam} (M)\leq D,-K\leq \sec(M)\leq K}$

The example of the lens chambers with shows that a lower limit for volume (or injectivity radius) cannot be dispensed with. ${\displaystyle S^{3}/\mathbb {Z} _{p}}$ ${\displaystyle \sec(M)\equiv 1,\operatorname {diam} (M)\leq 1}$

Individual evidence

1. ^ Jeff Cheeger: Comparison and finiteness theorems for Riemannian manifolds. Thesis (Ph.D.) - Princeton University. 1967
2. ^ Jeff Cheeger, David G. Ebin: Comparison theorems in Riemannian geometry. North-Holland Mathematical Library, Vol. 9. North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975.
3. ↑ Stefan Peters: Cheeger's finiteness theorem for diffeomorphism classes of Riemannian manifolds. J. Reine Angew. Math. 349: 77-82 (1984).
4. ↑ Michail Leonidowitsch Gromow : Metric structures for Riemannian and non-Riemannian spaces. Based on the 1981 French original. With appendices by M. Katz, P. Pansu and S. Semmes. Translated from the French by Sean Michael Bates. Progress in Mathematics, 152. Birkhauser Boston, Inc., Boston, MA, 1999. ISBN 0-8176-3898-9