Closure of the set of classical Riemannian spaces

Abstract

The basic result of the paper is a theorem asserting that the closure of the set of compact Riemannian spaces in the set of all compact metric spaces with inner metric consists precisely of the set of compact metric spaces with bilaterally bounded curvature in the sense of A. D. Aleksandrov. Here the convergence of a sequence of Riemannian spaces in the topology we consider means its Lipschitz convergence to a limit metric space and the uniform bilateral boundedness of the sectional curvatures of the spaces of the sequence. The results obtained are considered in application to the compactness theorem of M. Gromov.