# Geodesic Chord Property and Hypersurfaces of Space Forms

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

**Proposition**

**1.**

- 1.
- M satisfies the chord property.
- 2.
- The Gauss map G of M satisfies $G\left(x\right)=Ax+b$ for some $n\times n$ matrix A and a vector $b\in {\mathbb{E}}^{n}$.
- 3.
- M is an isoparametric hypersurface.
- 4.
- M is contained in one of the following hypersurfaces: ${\mathbb{E}}^{n-1}$, ${S}^{n-1}\left(r\right)$, ${S}^{p-1}\left(r\right)\times {\mathbb{E}}^{n-p}$.

## 2. Spherical Hypersurfaces

**Lemma**

**1.**

- 1.
- M satisfies the geodesic chord property.
- 2.
- For points $x,y\in M$, we have$${|A\left(x\right)}^{t}{y|=|A\left(y\right)}^{t}x|.$$
- 3.
- For points $x,y\in M$, we have$$\u2329G\left(x\right),y\u232a=\u03f5\u2329x,G\left(y\right)\u232a,$$

**Proof.**

**Remark**

**1.**

**Lemma**

**2.**

**Remark**

**2.**

**Proposition**

**2.**

**Theorem**

**1.**

- 1.
- M satisfies the geodesic chord property.
- 2.
- The Gauss map G satisfies $|\u2329G\left(x\right),y\u232a|=|\u2329G\left(y\right),x\u232a|$ for arbitrary $x,y\in M$.
- 3.
- The Gauss map G satisfies $G\left(x\right)=Ax+b$ for an $(n+1)\times (n+1)$ matrix A and a vector $b\in {\mathbb{E}}^{n+1}$.
- 4.
- M is an open portion of either a sphere ${S}^{n-1}\left(r\right)$ or a product ${S}^{p}\left({r}_{1}\right)\times {S}^{n-p-1}\left({r}_{2}\right)$ of spheres with ${r}_{1}^{2}+{r}_{2}^{2}=1$.

## 3. Hypersurfaces in the Hyperbolic Space

**Lemma**

**3.**

- 1.
- M satisfies the geodesic chord property.
- 2.
- For any two points $x,y\in M$, the frame matrix A of M satisfies$$\u2329A{\left(x\right)}^{t}\overline{y},A{\left(x\right)}^{t}\overline{y}\u232a=\u2329A{\left(y\right)}^{t}\overline{x},A{\left(y\right)}^{t}\overline{x}\u232a,$$
- 3.
- For any two points $x,y\in M$, the Gauss map $G\left(x\right)$ of M satisfies$${\u2329G\left(x\right),y\u232a}_{1}=\u03f5{\u2329x,G\left(y\right)\u232a}_{1},$$

**Lemma**

**4.**

**Case**

**1.**

**Case**

**2.**

**Case**

**3.**

**Proposition**

**3.**

- 1.
- ${S}^{n-1}(sinh\theta )\subset {H}^{n}$,
- 2.
- ${H}^{n-1}(cosh\theta )\subset {H}^{n}$,
- 3.
- ${S}^{p}(sinh\theta )\times {H}^{n-p-1}(cosh\theta )\subset {H}^{n}$,
- 4.
- $N=\left\{(x,f\left(x\right),f\left(x\right)+1)\right|f\left(x\right)=\frac{1}{2}{\left|x\right|}^{2},x\in {\mathbb{E}}^{n-1}\}\subset {H}^{n}$.

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

- 1.
- M satisfies the geodesic chord property.
- 2.
- The Gauss map G satisfies $|{\u2329G\left(x\right),y\u232a}_{1}|=|{\u2329G\left(y\right),x\u232a}_{1}|$ for any $x,y\in M$.
- 3.
- The Gauss map G satisfies $G\left(x\right)=Ax+b$ for an $(n+1)\times (n+1)$ matrix A and a vector $b\in {\mathbb{E}}_{1}^{n+1}$.
- 4.
- M is an isoparametric hypersurface of ${H}^{n}$.
- 5.
- M is an open part of one of the following hypersurfaces: ${S}^{n-1}\left(r\right)$, ${H}^{n-1}\left(r\right)$, ${S}^{p}\left({r}_{1}\right)\times {H}^{n-p-1}\left({r}_{2}\right)$, N, where ${r}_{2}^{2}-{r}_{1}^{2}=1$ and $N=\{(x,\frac{1}{2}{\left|x\right|}^{2},\frac{1}{2}{\left|x\right|}^{2}+1\left)\right|x\in {\mathbb{E}}^{n-1}\}$.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Rademacher, H.; Toeplitz, O. The Enjoyment of Mathematics; Translated from the Second (1933) German Edition and with Additional Chapters by H. Zuckerman; Princeton Science Library, Princeton University Press: Princeton, NJ, USA, 1994. [Google Scholar]
- Chen, B.-Y.; Kim, D.-S.; Kim, Y.H. New characterization of W-curves. Publ. Math. Debr.
**2006**, 69, 457–472. [Google Scholar] - Kim, D.-S.; Kim, Y.H. New characterizations of spheres, cylinders and W-curves. Linear Algebra Appl.
**2010**, 432, 3002–3006. [Google Scholar] [CrossRef] [Green Version] - Boas, H.P. A geometric characterization of the ball and the Bochner-Martinelli kernel. Math. Ann.
**1980**, 248, 275–278. [Google Scholar] [CrossRef] - Boas, H.P. Spheres and cylinders: A local geometric characterization. Ill. J. Math.
**1984**, 28, 120–124. [Google Scholar] [CrossRef] - Wegner, B. A differential geometric proof of the local geometric characterization of spheres and cylinders by Boas. Math. Balk. (N.S.)
**1988**, 2, 294–295. [Google Scholar] - Kim, D.-S. Ellipsoids and Elliptic hyperboloids in the Euclidean space E
^{n+1}. Linear Algebra Appl.**2015**, 471, 28–45. [Google Scholar] [CrossRef] - Kim, D.-S.; Kim, Y.H. Some characterizations of spheres and elliptic paraboloids. Linear Algebra Appl.
**2012**, 437, 113–120. [Google Scholar] [CrossRef] [Green Version] - Kim, D.-S.; Kim, Y.H. Some characterizations of spheres and elliptic paraboloids II. Linear Algebra Appl.
**2013**, 438, 1356–1364. [Google Scholar] [CrossRef] - Kim, D.-S.; Song, B. A characterization of elliptic hyperboloids. Honam Math. J.
**2013**, 35, 37–49. [Google Scholar] [CrossRef] - Kim, D.-S.; Kim, Y.H.; Yoon, D.W. On standard imbeddings of hyperbolic spaces in the Minkowski space. Comptes Rendus Math.
**2014**, 352, 1033–1038. [Google Scholar] [CrossRef] - Kim, D.-S. On the Gauss map of Hypersurfaces in the space form. J. Korean Math. Soc.
**1995**, 32, 509–518. [Google Scholar] - Kühnel, W. Differential Geometry, Curves-Surfaces-Manifolds; Translated from the 1999 German Original by Bruce Hunt; Student Mathematical Library, 16; American Mathematical Society: Providence, RI, USA, 2002. [Google Scholar]
- O’Neill, B. Semi-Riemannian Geometry with Applications to Relativity; Pure and Applied Mathematics, 103; Academic Press, Inc.: New York, NY, USA, 1983. [Google Scholar]

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kim, D.-S.; Kim, Y.H.; Yoon, D.W.
Geodesic Chord Property and Hypersurfaces of Space Forms. *Symmetry* **2019**, *11*, 1052.
https://doi.org/10.3390/sym11081052

**AMA Style**

Kim D-S, Kim YH, Yoon DW.
Geodesic Chord Property and Hypersurfaces of Space Forms. *Symmetry*. 2019; 11(8):1052.
https://doi.org/10.3390/sym11081052

**Chicago/Turabian Style**

Kim, Dong-Soo, Young Ho Kim, and Dae Won Yoon.
2019. "Geodesic Chord Property and Hypersurfaces of Space Forms" *Symmetry* 11, no. 8: 1052.
https://doi.org/10.3390/sym11081052