On translations in hyperbolic geometry of arbitrary (finite or infinite) dimension \(> 1\). (English) Zbl 1289.51015
Summary: Hyperbolic geometry of the plane was discovered by J. Bolyai (1802–1860), C. F. Gauß (1777–1855), and N. Lobachevski (1793–1856).
In our book [W. Benz, Classical geometries in modern contexts. Geometry of real inner product spaces. 3rd expanded ed., Basel: Birkhäuser (2012; Zbl 1253.51002)] we associate to every real vector space \(X\) of finite or infinite dimension \(> 1\), and equipped with a fixed inner product \(\delta : X \times X \to \mathbb R\), a hyperbolic geometry such that \((X,\delta)\), \((X',\delta')\) are isomorphic if, and only if, the associated hyperbolic geometries are isomorphic.
In this paper we present a common treatment of translations in Euclidean and hyperbolic geometry of arbitrary (finite or infinite) dimension greater than one.
In our book [W. Benz, Classical geometries in modern contexts. Geometry of real inner product spaces. 3rd expanded ed., Basel: Birkhäuser (2012; Zbl 1253.51002)] we associate to every real vector space \(X\) of finite or infinite dimension \(> 1\), and equipped with a fixed inner product \(\delta : X \times X \to \mathbb R\), a hyperbolic geometry such that \((X,\delta)\), \((X',\delta')\) are isomorphic if, and only if, the associated hyperbolic geometries are isomorphic.
In this paper we present a common treatment of translations in Euclidean and hyperbolic geometry of arbitrary (finite or infinite) dimension greater than one.
MSC:
| 51M10 | Hyperbolic and elliptic geometries (general) and generalizations |
