Lie Sphere Geometry in Hubert Spaces

Abstract

We develop Lie sphere geometry for arbitrary real pre-Hilbert spaces of (finite or infinite) dimension at least 2. One of the main results is that a bijection of the set of all Laguerre cycles which preserves contact in one direction must already be a Lie transformation (THEOREM 2). As a first consequence of this theorem we get that a bijection of an arbitrary real pre-Hilbert space of dimension at least 3 which preserves Lorentz-Minkowski distance 0 in one direction must already be a (proper or improper) Lorentz boost up to a dilatation, a translation and an orthogonal mapping (THEOREM 3). This is a generalization of results of A.D. Alexandroff [1], E.M. Schröder [21] and F. Cacciafesta [7]. Another consequence is that a bijection of the set of all Lie cycles which preserves contact in one direction must already be a Lie transformation (THEOREM 4). If we apply this result to the finite dimensional case, we get that the diffeomorphism assumption in the Fundamental Theorem of Lie sphere geometry as stated in Theorem 1.5 in T.E. Cecil [8], p. 33, is not needed for the proof of this theorem (REMARK to THEOREM 4).