On translations in hyperbolic geometry of arbitrary (finite or infinite) dimension \(>1\)

Abstract. 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 [3] we associate to every real vector space \(X\) of finite or infiinite 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) imension greater than one.

Key words and phrases. Hyperbolic geometry of arbitrary dimensional real inner product spaces, hyperbolic translations.