Closest point of the cut locus to submanifold
type: Item
instance of: scholarly article
title: Closest point of the cut locus to submanifold
zbMATH Open document ID: 0669.53040
DOI: 10.1007/BF02863634
author: Hukum Singh
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 1988-01-01T00:00:00Z
review text: Let M be a complete Riemannian manifold, L a compact connected submanifold of M and \(c_ 1,c_ 2: [0,\infty]\to M\) two unit speed geodesics of M emanating perpendicularly from L. Moreover, suppose that there exists an \(r>0\) such that \(x:=c_ 1(r)=c_ 2(r)\), but \(\dot c{}_ 1(r)\neq \dot c_ 2(r)\), where r is the distance from x to L; in particular, x is a cut point of L, but x is assumed not to be a focal point. The main result states that then in every neighbourhood of x there are cut points of L, which are nearer to L than x. For the proof the author derives a generalization of the Gauß-Lemma describing the relation of the geometry of the normal bundle of L and the geometry of M (via the normal exponential map).
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C22
zbMATH DE Number: 4096313
zbMATH Keywords: submanifold
zbMATH Keywords: cut points
zbMATH Keywords: Gauß-Lemma
zbMATH Keywords: normal exponential map
reviewed by: Helmut Reckziegel
MaRDI profile type: Publication
cites work: Q4169382
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full work available at URL: https://doi.org/10.1007/bf02863634
OpenAlex ID: W2060748765
description: scientific article; zbMATH DE number 4096313