Resumen de Möbius sphere geometry in inner product spaces

Walter Benz

• We develop Möbius sphere geometry for arbitrary euclidean spaces (i.e. real inner product spaces or real pre-Hilbert spaces) X of (finite or infinite) dimension at least 2. All Möbius transformations of X are determined, especially those which are involutorial. Moreover, M-transformations are characterized within the group of Lie transformations of X. We prove that the 4-point-invariants must be functions of the cross ratio. Stereographic projection from a hypersphere of $ X \oplus \mathbb{R} $ onto $ X \cup\{\infty\} $ is introduced, and also Poincarés model of hyperbolic geometry with respect to an M-ball B and one of the sides $ \Sigma $ of B. All bijections of $ \Sigma $ preserving hyperbolic distances are determined: they are exactly the Möbius transformations satisfying $ \mu(\Sigma) = \Sigma $. An isomorphism between the models of Poincaré and Weierstrass of hyperbolic geometry over X is established.