Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2304.02895 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912547535323136 |
|---|---|
| author | Grieser, Daniel Lye, Jørgen Olsen |
| author_facet | Grieser, Daniel Lye, Jørgen Olsen |
| contents | We study the behaviour of geodesics on a Riemannian manifold near a generalized conical or cuspidal singularity. We show that geodesics entering a small neighbourhood of the singularity either hit the singularity or approach it to a smallest distance $δ$ and then move away from it, winding around the singularity a number of times. We study the limiting behaviour $δ\to0$ in the second case. In the cuspidal case the number of windings goes to infinity as $δ\to0$, and we compute the precise asymptotic behaviour of this number. The asymptotics have explicitly given leading term determined by the warping factor that describes the type of cuspidal singularity. We also discuss in some detail the relation between differential and metric notions of conical and cuspidal singularities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_02895 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Geodesics Orbiting a Singularity Grieser, Daniel Lye, Jørgen Olsen Differential Geometry 53C22 (Primary) 53A99, 53D25 (Secondary) We study the behaviour of geodesics on a Riemannian manifold near a generalized conical or cuspidal singularity. We show that geodesics entering a small neighbourhood of the singularity either hit the singularity or approach it to a smallest distance $δ$ and then move away from it, winding around the singularity a number of times. We study the limiting behaviour $δ\to0$ in the second case. In the cuspidal case the number of windings goes to infinity as $δ\to0$, and we compute the precise asymptotic behaviour of this number. The asymptotics have explicitly given leading term determined by the warping factor that describes the type of cuspidal singularity. We also discuss in some detail the relation between differential and metric notions of conical and cuspidal singularities. |
| title | Geodesics Orbiting a Singularity |
| topic | Differential Geometry 53C22 (Primary) 53A99, 53D25 (Secondary) |
| url | https://arxiv.org/abs/2304.02895 |