Removing point singularities of Riemannian manifolds

Smith, P. D.; Yang, Deane

doi:10.1090/S0002-9947-1992-1052910-2

Removing point singularities of Riemannian manifolds
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by P. D. Smith and Deane Yang PDF

Trans. Amer. Math. Soc. 333 (1992), 203-219 Request permission

Abstract:

We study the behavior of geodesics passing through a point singularity of a Riemannian manifold. In particular, we show that if the curvature does not blow up too rapidly near the singularity, then the singularity is at worst an orbifold singularity. The idea is to construct the exponential map centered at a singularity. Since there is no tangent space at the singularity, a surrogate is needed. We show that the vector space of radially parallel vector fields is well defined and that there is a correspondence between unit radially parallel vector fields and geodesics emanating from the singular point.

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