The cut locus on low-dimensional noncompact manifolds
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.
scientific article; zbMATH DE number 4209013