Cut loci, minimizers, and wavefronts in Riemannian manifolds with boundary
type: Item
instance of: scholarly article
title: Cut loci, minimizers, and wavefronts in Riemannian manifolds with boundary
zbMATH Open document ID: 0817.53023
author: Stephanie B. Alexander
author: I. David Berg
author: Richard L. Bishop
published in: Michigan Mathematical Journal
publication date: 1995-08-03T00:00:00Z
review text: The notion of cut loci in manifolds without boundary was discussed by S. Kobayashi, F. W. Warner and A. D. Weinstein [see \textit{S. Kobayashi} and \textit{K. Nomizu}, Foundations of differential geometry. Vol. II (1969; Zbl 0175.485)]. \textit{F.-E. Wolter} [Arch. Math. 32, 92-96 (1979; Zbl 0409.53032); Cut loci in bordered and unbordered Riemannian manifolds, Dissertation, Techn. Univ. Berlin (1985; Zbl 0674.53049)] studied cut locus in Riemannian manifolds with boundary. In fact, Wolter revised the definition of cut locus in Riemannian manifolds with boundary, considered several variants thereof and analyzed the relationship among them. In the present note the authors consider a connected, metrically complete \(C^ \infty\)-Riemannian manifold with boundary, and in the light of the fact that minimizers can bifurcate and merge, there are new features which the authors illustrate by some examples. There may exist open sets each of whose points \(p\) has the property that every point \(q\) sufficiently far from \(p\) has more than one minimizer from \(p\). The authors show that for any \(p\) and almost all \(q\), there is a natural way to choose exactly one minimizer from \(p\) to \(q\). The authors also investigate the ways in which the cut locus of the definition (the cut locus of \(p\) is the closure of the set of points having two minimizers from \(p\) with different terminal velocities) diverges from the classical case. The main result of the paper is as follows: For any given point \(p\), let \(M^*(p)\) consist of the points \(q\) all of whose minimizers from \(p\) have the same terminal velocity vector. Then the complement of \(M^*(p)\) has measure zero, and \(M^*(p)\) is the union of the unique primary minimizers from \(p\) to each of its points. The traces \(pq\) of these minimizers satisfy the branching condition \(pq_ 1 \cap pq_ 2 = pq_ 3\) for some \(q_ 3\).
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 474968
zbMATH Keywords: cut locus
Wikidata QID: Q115238980
reviewed by: Om Prakash Singh
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029004750
OpenAlex ID: W2066230043
DOI: 10.1307/MMJ/1029004750
Recommended article: Q3828675
Recommended article: Q3799275
Recommended article: On the cut locus of closed sets
Recommended article: Q3763216
Recommended article: Remarks on manifolds with two-sided curvature bounds
Recommended article: Laplacian of the distance function on the cut locus on a Riemannian manifold
Recommended article: On the terminal points of co-rays and rays
Recommended article: Q3706130
Recommended article: Q5447636
Recommended article: Some Consequences of the Nature of the Distance Function on the Cut Locus in a Riemannian Manifold
description: scientific article; zbMATH DE number 474968