Desingularization of Singular Riemannian Foliation

Let F be a singular Riemannian foliation on a compact Riemannian manifold M . By successive blow-ups along the strata of F we construct a regular Riemannian foliation F̂ on a compact Riemannian manifold M̂ and a desingularization map ρ̂ : M̂ → M that projects leaves of F̂ into leaves of F . This result generalizes a previous result due to Molino for the particular case of a singular Riemannian foliation whose leaves were the closure of leaves of a regular Riemannian foliation. We also prove that, if the leaves of F are compact, then, for each small ǫ > 0, we can find M̂ and F̂ so that the desingularization map induces an ǫ-isometry between M/F and M̂/F̂ . This implies in particular that the space of leaves M/F is a Gromov-Hausdorff limit of a sequence of Riemannian orbifolds {(M̂n/F̂n)}.