### Abstract

With the aim of revealing their purely geometric nature, we rephrase two theorems of S. Yang and A. Fang [S. Yang, A. Fang, A new characteristic of Möbius transformations in hyperbolic geometry, J. Math. Anal. Appl. 319 (2006) 660-664] characterizing Möbius transformations as definability results in elementary plane hyperbolic geometry. We show not only that elementary plane hyperbolic geometry can be axiomatized in terms of the quaternary predicates λ or σ, with λ (a b c d) to be read as 'a b c d is a Lambert quadrilateral' and σ (a b c d) to be read as 'a b c d is a Saccheri quadrilateral', but also that all elementary notions of hyperbolic geometry can be positively defined (i.e. by using only quantifiers (∀ and ∃) and the connectives ∨ and ∧ in the definiens) in terms of λ or σ.

Original language | English (US) |
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Pages (from-to) | 531-532 |

Number of pages | 2 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 346 |

Issue number | 2 |

DOIs | |

State | Published - Oct 15 2008 |

Externally published | Yes |

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### Keywords

- Hyperbolic geometry
- Lambert quadrilaterals
- Möbius transformations
- Saccheri quadrilaterals

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics