### Abstract

It is well known that the helicoids are the only ruled minimal surfaces in ℝ^{3}. The similar characterization for ruled minimal surfaces can be given in many other 3-dimensional homogeneous spaces. In this note we consider the product space M × ℝ for a 2-dimensional manifold M and prove that M ×ℝ has a nontrivial minimal surface ruled by horizontal geodesics only when M has a Clairaut parametrization. Moreover such minimal surface is the trace of the longitude rotating in M while translating vertically in constant speed in the direction of ℝ.

Original language | English |
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Pages (from-to) | 1887-1892 |

Number of pages | 6 |

Journal | Bulletin of the Korean Mathematical Society |

Volume | 53 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2016 |

### Keywords

- Helicoid
- Minimal surface
- Ruled surface

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Jin, Y., Kim, Y. W., Park, N., & Shin, H. (2016). Ruled minimal surfaces in product spaces.

*Bulletin of the Korean Mathematical Society*,*53*(6), 1887-1892. https://doi.org/10.4134/BKMS.b160006