Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

The Lipschitz continuity of the distance function to the cut locus
HTML articles powered by AMS MathViewer

by Jin-ichi Itoh and Minoru Tanaka PDF
Trans. Amer. Math. Soc. 353 (2001), 21-40 Request permission

Abstract:

Let $N$ be a closed submanifold of a complete smooth Riemannian manifold $M$ and $U\nu$ the total space of the unit normal bundle of $N$. For each $v \in U\nu$, 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 $U\nu$ 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 $U\nu$. 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.
References
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 53C22, 28A78
  • Retrieve articles in all journals with MSC (2000): 53C22, 28A78
Additional Information
  • Jin-ichi Itoh
  • Affiliation: Faculty of Education, Kumamoto University, Kumamoto 860-8555 Japan
  • MR Author ID: 212444
  • 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
  • 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
  • © Copyright 2000 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 353 (2001), 21-40
  • MSC (2000): Primary 53C22; Secondary 28A78
  • DOI: https://doi.org/10.1090/S0002-9947-00-02564-2
  • MathSciNet review: 1695025