The cut locus on low-dimensional noncompact manifolds
type: Item
instance of: scholarly article
title: The cut locus on low-dimensional noncompact manifolds
zbMATH Open document ID: 0731.53045
DOI: 10.1007/BF02411356
author: Bruce L. Reinhart
published in: Annals of Global Analysis and Geometry
publication date: 1991-01-01T00:00:00Z
review text: Let \(M\) be a connected smooth complete Riemannian \(n\)-manifold, \(P\in M\), let \(C\) be the cut locus of \(P\). \(\hat M\) is the compactification of \(M\) obtained by adding one point to each end of \(M\), \(\hat C\) is the closure of \(C\) in \(M\). Let \(D^n\) be the \(n\)-dimensional disk and let \(S^{n-1}\) be its boundary. It is shown that there is a map \(\varepsilon: (D^n,S^{n-1})\to (\hat M,\hat C)\) equal to the exponential map near the origin which induces a homeomorphism of the interior of \(D^n\) onto \(M-C\). \(\varepsilon\) is constructed by an appropriate rescaling of the exponential map in the radial directions. Furthermore, for two base points \(P_1\) and \(P_2\) and two complete Riemannian metrics the corresponding maps \(\varepsilon_j\) can be joined by a path of such maps.
Mathematics Subject Classification ID: 53C22
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 4209013
zbMATH Keywords: cut locus
zbMATH Keywords: end
zbMATH Keywords: exponential map
reviewed by: Christian Bär
MaRDI profile type: Publication
cites work: Stability of the cut locus in dimensions less than or equal to 6
cites work: Q4194161
cites work: On complete open manifolds of positive curvature
full work available at URL: https://doi.org/10.1007/bf02411356
OpenAlex ID: W2044781288
description: scientific article; zbMATH DE number 4209013