Cheeger finiteness theorem

A theorem stating that for given positive numbers $n$, $d$, $v$, $\kappa$ there exist only finitely many diffeomorphism classes of compact $n$-dimensional Riemannian manifolds $M$ satisfying

\begin{equation*} \operatorname{diam}M \leq d, \end{equation*}

\begin{equation*} \operatorname{Vol}( M ) \leq v , | \text { sec. curv. } M | \leq \kappa, \end{equation*}

i.e. for every given sequence of compact $n$-dimensional Riemannian manifolds satisfying these bounds, there is an infinite subsequence for which any two of the manifolds are diffeomorphic.

The proof is based on discretizations of Riemannian manifolds and on lower bounds for the injectivity radius (cf. Berger inequality) in terms of $n$, $d$, $v$, $\kappa$.

Cf. also Riemannian manifold.

References

[a1]	I. Chavel, "Riemannian geometry: A modern introduction" , Cambridge Univ. Press (1995)
[a2]	J. Cheeger, "Finiteness theorems in Riemannian manifolds" Amer. J. Math. , 92 (1970) pp. 61–74