Author:
Grieser Daniel,Lye Jørgen Olsen
Abstract
AbstractWe 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.
Funder
Deutsche Forschungsgemeinschaft
Gottfried Wilhelm Leibniz Universität Hannover
Publisher
Springer Science and Business Media LLC
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