Mappings preserving the area equality of hyperbolic triangles are motions. (English) Zbl 1213.51017
Let \(H\) be the complete hyperbolic plane.
In a series of papers by the author, the theme is characterization of isometry on \(H\).
In this paper he considers the quaternary relation \(\Delta(abcd)\), where \(a,b,c,d\in H\) and \(\triangle abc=\triangle abd\), (they have the same hyperbolic area).
The conclusion is that if \(\varphi:H\to H\) preserves the quaternary relation, that is, \(\Delta(abcd) \Rightarrow \Delta(\varphi(a)\varphi(b)\varphi(c)\varphi(d))\) for any \(a,b,c,d\in H\), then \(\varphi\) is an isometry. He has faith that he doesn’t use trigonometric functions but purely geometric and abundant theorems on his theme.
In a series of papers by the author, the theme is characterization of isometry on \(H\).
In this paper he considers the quaternary relation \(\Delta(abcd)\), where \(a,b,c,d\in H\) and \(\triangle abc=\triangle abd\), (they have the same hyperbolic area).
The conclusion is that if \(\varphi:H\to H\) preserves the quaternary relation, that is, \(\Delta(abcd) \Rightarrow \Delta(\varphi(a)\varphi(b)\varphi(c)\varphi(d))\) for any \(a,b,c,d\in H\), then \(\varphi\) is an isometry. He has faith that he doesn’t use trigonometric functions but purely geometric and abundant theorems on his theme.
Reviewer: Kazushi Ahara (Kawasaki)
MSC:
| 51M10 | Hyperbolic and elliptic geometries (general) and generalizations |
| 51M25 | Length, area and volume in real or complex geometry |
| 03C40 | Interpolation, preservation, definability |
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