Finiteness theorem

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

Statement

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

$dim(M)=n,|K|\leq \Lambda ,diam(M)\leq D,vol(M)\geq v$

where $K$ denotes the sectional curvature.

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