Cheeger-Colding theory

Cheeger-Colding theory is a body of results in metric geometry and geometric analysis that establishes how geometric and analytic properties of Riemannian manifolds persist under measured Gromov-Hausdorff limits. Developed by Jeff Cheeger and Tobias Colding in the 1990s, the theory shows that manifolds with uniform lower Ricci curvature bounds converge to limit spaces that — despite potentially being singular — retain a remarkable amount of structure: they admit a well-defined dimension, satisfy sharp volume comparison theorems, and support a first-order differential calculus through the theory of Alexandrov spaces. The theory was instrumental in proving that the fundamental group of a manifold with nonnegative Ricci curvature is almost nilpotent, and it provides the analytic foundation for synthetic Ricci curvature by demonstrating that curvature-controlled phenomena are metric rather than smooth in essence.