### Abstract

We show that any finite graph without loops can be realized as the Reeb graph of a smooth function on a closed manifold with finitely many critical values, but possibly with positive dimensional critical point set. We also show that such a function can be chosen as the height function on a surface immersed in 3-space, provided that the graph has no isolated vertices.

Original language | English |
---|---|

Pages (from-to) | 75-84 |

Number of pages | 10 |

Journal | Kyushu Journal of Mathematics |

Volume | 65 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jun 10 2011 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Kyushu Journal of Mathematics*,

*65*(1), 75-84. https://doi.org/10.2206/kyushujm.65.75

**A smooth function on a manifold with given reeb graph.** / Masumoto, Yasutaka; Saeki, Osamu.

Research output: Contribution to journal › Article

*Kyushu Journal of Mathematics*, vol. 65, no. 1, pp. 75-84. https://doi.org/10.2206/kyushujm.65.75

}

TY - JOUR

T1 - A smooth function on a manifold with given reeb graph

AU - Masumoto, Yasutaka

AU - Saeki, Osamu

PY - 2011/6/10

Y1 - 2011/6/10

N2 - We show that any finite graph without loops can be realized as the Reeb graph of a smooth function on a closed manifold with finitely many critical values, but possibly with positive dimensional critical point set. We also show that such a function can be chosen as the height function on a surface immersed in 3-space, provided that the graph has no isolated vertices.

AB - We show that any finite graph without loops can be realized as the Reeb graph of a smooth function on a closed manifold with finitely many critical values, but possibly with positive dimensional critical point set. We also show that such a function can be chosen as the height function on a surface immersed in 3-space, provided that the graph has no isolated vertices.

UR - http://www.scopus.com/inward/record.url?scp=79958056122&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79958056122&partnerID=8YFLogxK

U2 - 10.2206/kyushujm.65.75

DO - 10.2206/kyushujm.65.75

M3 - Article

VL - 65

SP - 75

EP - 84

JO - Kyushu Journal of Mathematics

JF - Kyushu Journal of Mathematics

SN - 1340-6116

IS - 1

ER -