Year: 2023
Type: article
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 $$\delta $$ δ and then move away from it, winding around the singularity a number of times. We study the limiting behaviour $$\delta \rightarrow 0$$ δ → 0 in the second case. In the cuspidal case the number of windings goes to infinity as $$\delta \rightarrow 0$$ 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.
Source: Journal of Geometry
Authors: Daniel Grieser, Jørgen Olsen Lye
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,890
OpenAlex ID: W4389505707