## The Lipschitz continuity of the distance function to the cut locus

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- 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

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## 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