Finiteness theorem

This article describes a result related to the sectional curvature of a Riemannian manifold

Statement

Let $n\in \mathbb{N}$, $\mathrm{\Lambda }<\mathrm{\infty }$, $D<\mathrm{\infty }$m and $v>0$. Then consider the space $S$ of differential manifolds $M$ admitting a metric $g$ for which:

$dim(M)=n,\left|K\right|\le \mathrm{\Lambda },diam(M)\le D,vol(M)\ge v$

where $K$ denotes the sectional curvature.

Then, $S$ consists of finitely many diffeomorphism classes/types.