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

TL;DR

This paper establishes a broad collapsing theorem for stratified Riemannian metrics on orbifolds, allowing for lower-dimensional Gromov-Hausdorff limits without curvature bounds, and introduces new geometric structures and fibrations on orbifolds.

Contribution

It presents a novel collapsing result that does not require curvature or injectivity radius bounds and introduces new classes of stratified geometric structures and fibrations on orbifolds.

Findings

Proves a collapsing theorem for stratified Riemannian metrics on orbifolds.

Introduces weak submersions and new stratified geometric structures.

Allows Gromov-Hausdorff limits with lower dimension than the original orbifold.

Abstract

This paper proves a general collapsing result for families of stratified Riemannian metrics $\widehat{g}^\mu$ on a compact orbifold $E$, subject to suitable limiting conditions on the metrics $\widehat{g}^\mu$ 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 $\left(E,\widehat{g}^\mu\right)$ and thus allows for Gromov-Hausdorff limits of $\left(E,\widehat{g}^\mu\right)$ 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.