The construction of \(\epsilon\)-splitting map (Q2677321)

The main result of the paper is Theorem 4.9 that is a construction of the \(\epsilon\)-splitting map from a concentric geodesic ball (in a complete Riemannian manifold with non-negative Ricci curvature) with uniformly small radius to the corresponding Euclidean ball. The existence and the application of \(\epsilon\)-splitting maps were showed by Cheeger in [\textit{J. Cheeger}, Degeneration of Riemannian metrics under Ricci curvature bounds. Pisa: Scuola Normale Superiore; Rome: Accademia Nazionale dei Lincei (2001; Zbl 1055.53024)]. We note that the existence of the \(\epsilon\)-splitting map is equivalent to the existence of an \(\tilde{\epsilon}\)-Gromov-Hausdorff approximation between such two spaces.