A general collapsing result for families of stratified Riemannian metrics on orbifolds
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911375197995008 |
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| author | Mayther, Laurence H. |
| author_facet | Mayther, Laurence H. |
| contents | This paper proves a general collapsing result for families of stratified Riemannian metrics $\widehat{g}^μ$ on a compact orbifold $E$, subject to suitable limiting conditions on the metrics $\widehat{g}^μ$ as $μ\to \infty$. The result is distinct from similar theorems in the literature since it does not require bounds on curvature or injectivity radius of $\left(E,\widehat{g}^μ\right)$ and thus allows for Gromov-Hausdorff limits of $\left(E,\widehat{g}^μ\right)$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_05016 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A general collapsing result for families of stratified Riemannian metrics on orbifolds Mayther, Laurence H. Differential Geometry Geometric Topology Metric Geometry 53C23, 53C15, 53C60, 57R18, 58A35 (Primary) 28A20, 28A75 (Secondary) This paper proves a general collapsing result for families of stratified Riemannian metrics $\widehat{g}^μ$ on a compact orbifold $E$, subject to suitable limiting conditions on the metrics $\widehat{g}^μ$ as $μ\to \infty$. The result is distinct from similar theorems in the literature since it does not require bounds on curvature or injectivity radius of $\left(E,\widehat{g}^μ\right)$ and thus allows for Gromov-Hausdorff limits of $\left(E,\widehat{g}^μ\right)$ 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. |
| title | A general collapsing result for families of stratified Riemannian metrics on orbifolds |
| topic | Differential Geometry Geometric Topology Metric Geometry 53C23, 53C15, 53C60, 57R18, 58A35 (Primary) 28A20, 28A75 (Secondary) |
| url | https://arxiv.org/abs/2308.05016 |