Cheeger's Finiteness Theorem

Consider the set of compact $n$-Riemannian Manifolds $M$ with diameter$(M)\leq d$, Volume$(M)\geq V$, and $\vert{\mathcal K}\vert\leq \kappa$ where $\kappa$ is the Sectional Curvature. Then there is a bound on the number of Diffeomorphisms classes of this set in terms of the constants $n$, $d$, $V$, and $\kappa$.

References

Chavel, I. Riemannian Geometry: A Modern Introduction. New York: Cambridge University Press, 1994.