It is a work in progress with G. Besson, G. Courtois et A. Sambusetti.

From a (1 + ε, η)-quasi-isometry between two CAT(1) Riemannian manifolds of the same dimension (the second one having moreover Ricci curvature bounded from below), we explicitely construct a local diffeomorphism. This provides results of differential rigidity for the canonical structures of Rⁿ and Sⁿ.

For two locally CAT(1) compact Riemannian manifolds whose mutual Gromov-Hausdorff distance is < ε, it also gives a condition to be diffeomorphic. The (positive) values of ε and η are computable.