Lambert or Saccheri quadrilaterals as single primitive notions for plane hyperbolic geometry

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:346

Article

Pambuccian, Victor

关键词: Mobius transformations;    Lambert quadrilaterals;    Saccheri quadrilaterals;    hyperbolic geometry;   

DOI  :  10.1016/j.jmaa.2008.05.076

来源: Elsevier

【 摘 要 】

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 Mobius transformations in hyperbolic geometry, J. Math. Anal. Appl. 319 (2006) 660-664] characterizing Mobius 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 lambda or sigma, with lambda(abcd) to be read as 'abcd is a Lambert quadrilateral' and sigma(abcd) to be read as 'abcd is a Saccheri quadrilateral', but also that all elementary notions of hyperbolic geometry can be positively defined (i.e. by using only quantifiers (for all and there exists) and the connectives boolean OR and boolean AND in the definiens) in terms of lambda or sigma. (C) 2008 Elsevier Inc. All rights reserved.

【 授权许可】

Free