Gromov's Compactness Theorem states that a sequence of Riemannian manifolds with uniform bounded curvature and diameter will have a subsequence that converges in the Gromov-Hausdorff sense to a limit space, which can be a singular space or a manifold. This theorem provides essential insights into the behavior of geometric structures under specific constraints, leading to conclusions about the topology and geometry of the limit space. It plays a critical role in understanding the compactness properties of manifolds with bounded curvature.
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