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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The Lipschitz continuity of the distance function to the cut locus


Authors: Jin-ichi Itoh and Minoru Tanaka
Journal: Trans. Amer. Math. Soc. 353 (2001), 21-40
MSC (2000): Primary 53C22; Secondary 28A78
Published electronically: August 3, 2000
MathSciNet review: 1695025
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Abstract:

Let $N$ be a closed submanifold of a complete smooth Riemannian manifold $M$ and $U\mbox{{$\nu$ }}$ the total space of the unit normal bundle of $N$. For each $v \in U\mbox{{$\nu$ }}$, let $\rho(v) $ denote the distance from $N$ to the cut point of $N$ on the geodesic $\gamma_v$ with the velocity vector $\dot\gamma_v(0)=v.$ The continuity of the function $\rho$ on $U\mbox{{$\nu$ }}$ is well known. In this paper we prove that $\rho$ is locally Lipschitz on which $\rho$is bounded; in particular, if $M$ and $N$ are compact, then $\rho$ is globally Lipschitz on $U\mbox{{$\nu$ }}$. Therefore, the canonical interior metric $\delta$ may be introduced on each connected component of the cut locus of $N,$ and this metric space becomes a locally compact and complete length space.


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Additional Information

Jin-ichi Itoh
Affiliation: Faculty of Education, Kumamoto University, Kumamoto 860-8555 Japan
Email: j-itoh@gpo.kumamoto-u.ac.jp

Minoru Tanaka
Affiliation: Department of Mathematics, Tokai University, Hiratsuka 259-1292, Japan
Email: m-tanaka@sm.u-tokai.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02564-2
PII: S 0002-9947(00)02564-2
Received by editor(s): October 14, 1998
Received by editor(s) in revised form: April 13, 1999
Published electronically: August 3, 2000
Additional Notes: Supported in part by a Grant-in-Aid for Scientific Research from The Ministry of Education, Science, Sports and Culture, Japan
Article copyright: © Copyright 2000 American Mathematical Society