Desingularization of singular Riemannian foliation

Abstract

Let ${\mathcal{F}}$ be a singular Riemannian foliation on a compact Riemannian manifold M. By successive blow-ups along the strata of ${\mathcal{F}}$ we construct a regular Riemannian foliation ${\hat{\mathcal{F}}}$ on a compact Riemannian manifold ${\hat{M}}$ and a desingularization map ${\hat{\rho}:\hat{M}\rightarrow M}$ that projects leaves of ${\hat{\mathcal{F}}}$ into leaves of ${\mathcal{F}}$. 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 ${\mathcal{F}}$ are compact, then, for each small ${\epsilon >0 }$, we can find ${\hat{M}}$ and ${\hat{\mathcal{F}}}$ so that the desingularization map induces an ${\epsilon}$-isometry between ${M/\mathcal{F}}$ and ${\hat{M}/\hat{\mathcal{F}}}$. This implies in particular that the space of leaves ${M/\mathcal{F}}$ is a Gromov-Hausdorff limit of a sequence of Riemannian orbifolds ${\{(\hat{M}_{n}/\hat{\mathcal{F}}_{n})\}}$.