A general collapsing result for families of stratified Riemannian metrics on orbifolds

Abstract

This paper proves a general collapsing result for families of stratified Riemannian metrics ${\hat{g}}^{\mu }$ on a compact orbifold $E$, subject to suitable limiting conditions on the metrics ${\hat{g}}^{\mu }$ as $\mu \to \infty$. The result is distinct from similar theorems in the literature since it does not require bounds on curvature or injectivity radius of $\left(E,{\hat{g}}^{\mu }\right)$ and thus allows for Gromov-Hausdorff limits of $\left(E,{\hat{g}}^{\mu }\right)$ which have strictly lower dimension than $E$. The paper also introduces and studies a new class of stratified fibrations between orbifolds, termed weak submersions, and new classes of geometric structures on orbifolds, termed stratified Riemannian metrics, stratified Riemannian semi-metrics and stratified quasi-Finslerian structures, all of which play a key role in the proof of the main theorem.

arXiv comment29 pages; some minor typos corrected and contact details updated