Details
| Original language | English |
|---|---|
| Article number | 1 |
| Journal | Journal of Geometry |
| Volume | 115 |
| Issue number | 1 |
| Publication status | Published - 9 Dec 2023 |
Abstract
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→ 0 in the second case. In the cuspidal case the number of windings goes to infinity as δ→ 0 , 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.
Keywords
- Conical singularity, Cuspidal singularity, Geodesics, Singular hamiltonian systems
ASJC Scopus subject areas
- Mathematics(all)
- Geometry and Topology
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In: Journal of Geometry, Vol. 115, No. 1, 1, 09.12.2023.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Geodesics orbiting a singularity
AU - Grieser, Daniel
AU - Lye, Jørgen Olsen
N1 - Funding Information: Open Access funding enabled and organized by Projekt DEAL. The first author was partially supported by DFG Priority Programme 2026 ‘Geometry at Infinity’. The second author did not receive any grant for writing this article.
PY - 2023/12/9
Y1 - 2023/12/9
N2 - 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→ 0 in the second case. In the cuspidal case the number of windings goes to infinity as δ→ 0 , 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.
AB - 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→ 0 in the second case. In the cuspidal case the number of windings goes to infinity as δ→ 0 , 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.
KW - Conical singularity
KW - Cuspidal singularity
KW - Geodesics
KW - Singular hamiltonian systems
UR - http://www.scopus.com/inward/record.url?scp=85179330173&partnerID=8YFLogxK
U2 - 10.48550/arXiv.2304.02895
DO - 10.48550/arXiv.2304.02895
M3 - Article
AN - SCOPUS:85179330173
VL - 115
JO - Journal of Geometry
JF - Journal of Geometry
SN - 0047-2468
IS - 1
M1 - 1
ER -