Metric curvature, convergence, and topological finiteness (Q1196381)

The author uses metric techniques to generalize the Gromov convergence theorem and the Cheeger finiteness theorem by replacing an upper curvature bound and lower volume bound with a lower bound on injectivity radius and by removing smoothness assumptions as well. In a first theorem the author obtains that an \(n\)-dimensional, metrically and geodesically complete inner metric space of curvature bounded below by some constant and injectivity radius bounded below by some positive constant has a \(C^ 1\) structure.