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Lie sphere geometry in Hilbert spaces. (English) Zbl 0995.51003
The classical \(n\)-dimensional Lie geometry, \(n\geq 2\), is the geometry of Lie cycles, that is, points and oriented hyperspheres of the \(n\)-sphere, with respect to oriented contact between them. Deleting a point of an \(n\)-dimensional Lie geometry one obtains the \(n\)-dimensional Laguerre geometry of points and oriented hyperspheres (the Laguerre cycles) and oriented hyperplanes (the Laguerre spears) of the \(n\)-dimensional Euclidean space. These kinds of geometries and their associated transformations have been studied by many authors and are of interest in relativity and differential geometry. In particular, the characterisation of geometric mappings under as few conditions as possible, especially without differentiability of continuity assumptions, is a longstanding interest of the present author.
In the paper under review the author develops Laguerre and Lie geometry dimension- and basis-free but close to the classical case for an arbitrary real pre-Hilbert space \(X\) of (finite or infinite) dimension at least 2 and determines the form of all Lie transformations. The author begins with the Laguerre geometry of \(X\) and introduces cycle coordinates (elements of \(X\oplus{\mathbb R}\)) and homogeneous spear coordinates (elements of the projective space over \(X\oplus{\mathbb R}\oplus{\mathbb R}\)). Oriented contact between Laguerre objects (spears or cycles) is then expressed using these coordinates and algebraic descriptions of various pencils and bundles of cycles are developed. In particular, the coordinates of touching cycles have distance 0 in the Lorentz-Minkowski distance on \(X\oplus{\mathbb R}\). The author then investigates Laguerre transformations, that is, permutations \(\lambda\) of the set of Laguerre objects that take spears to spears and cycles to cycles such that a spear and a cycle are in oriented contact if and only if their images under \(\lambda\) are in oriented contact. Identifying a cycle with all the spears it is in contact with, and likewise for spears, it is shown that Laguerre transformations can be define solely in terms of spears or cycles. Each Laguerre transformation is the composition of an orthogonal mapping, a Lorentz boost, a dilatation and a translation. Furthermore, a permutation of the Laguerre cycles that takes touching cycles to touching cycles must already come from a Laguerre transformation. As a consequence, the author obtains a complete description of the permutations of \(X\oplus{\mathbb R}\) that preserve Lorentz-Minkowski distance 0. The Laguerre geometry on \(X\) is extended to obtain the Lie geometry on \(X\). The Lie cycles are the Laguerre cycles and spears and a new symbol \(\infty\) and contact between Lie cycles naturally extends the contact relation in the Laguerre geometry. Lie transformations are characterised as those permutations of the Lie cycles that preserve contact in one direction. Since the stabilizer of \(\infty\) consists of all Laguerre transformations an explicit description of Lie transformations is obtained.

51B25 Lie geometries in nonlinear incidence geometry
53A30 Conformal differential geometry (MSC2010)
83A05 Special relativity
Full Text: DOI
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