These theorems have been widely used in the fields of geometric group theory and Riemannian geometry.
Gromov found a condition on a sequence of compact metric spaces which ensures that a subsequence converges to some metric space relative to the Gromov–Hausdorff distance:[1] Let (Xi, di) be a sequence of compact metric spaces with uniformly bounded diameter.
[2] Gromov first formally introduced it in his 1981 resolution of the Milnor–Wolf conjecture in the field of geometric group theory, where he applied it to define the asymptotic cone of certain metric spaces.
[4] Specializing to the setting of geodesically complete Riemannian manifolds with a fixed lower bound on the Ricci curvature, the crucial covering condition in Gromov's metric compactness theorem is automatically satisfied as a corollary of the Bishop–Gromov volume comparison theorem.
[6] This special case of the metric compactness theorem is significant in the field of Riemannian geometry, as it isolates the purely metric consequences of lower Ricci curvature bounds.