×

zbMATH — the first resource for mathematics

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.

MSC:
51B25 Lie geometries in nonlinear incidence geometry
53A30 Conformal differential geometry (MSC2010)
83A05 Special relativity
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alexandrov, A.D.: Seminar Report. Uspehi Mat. Nauk. 5 (1950), no. 3 (37), 187
[2] Benz, W.: Vorlesungen über Geometrie der Algebren. Die Grundlehren der math. Wissensch. in Einzeldarstellungen, Bd. 197, Springer-Verlag, Berlin, New York, 1973 · Zbl 0258.50024
[3] Benz, W: Geometrische Transformationen. BI Wissenschaftsverlag, Mannheim, Leipzig, Wien, Zürich, 1992.
[4] Benz, W: Lorentz-Minkowski distances in Hilbert spaces. Geom. Dedicata 81 (2000) 219–230 · Zbl 0959.51013 · doi:10.1023/A:1005235923736
[5] Benz, W: Real Geometries. BI Wissenschaftsverlag, Mannheim, Leipzig, Wien, Zürich, 1994.
[6] Blaschke, W: Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. III. Differentialgeometrie der Kreise und Kugeln. Bearbeitet von G. Thomsen. Grundlehren der math. Wissensch. in Einzeldarstellungen, Bd. 29, Springer-Verlag, Berlin, 1929 · JFM 55.0422.01
[7] Cacciafesta, F.: An observation about a theorem of A.D. Alexandrov concerning Lorentz transformations. Journ. Geom. 18 (1982) 5–8 · Zbl 0485.51015 · doi:10.1007/BF01947634
[8] Cecil, T.E.: Lie sphere geometry. Springer-Verlag, New York, Berlin, 1992 · Zbl 0752.53003
[9] Cecil, T.E. and Chern, S.S.: Tautness and Lie sphere geometry. Math. Ann. 278 (1987) 381–399 · Zbl 0635.53029 · doi:10.1007/BF01458076
[10] Chen, Y.: Zur Axiomatik der ebenen Geometrie von Lie über einem Körper. Dissertation Bochum, 1967
[11] Dubikajtis, L.: La géométrie de Lie. Rozprawy Mat. 15 (1958) · Zbl 0081.37104
[12] Höfer, R.: On the geometry of hyperbolic cycles. Aequat. Math. 56 (1998) 169–180 · Zbl 0917.51020 · doi:10.1007/s000100050053
[13] Höfer, R.: Cycles in de Sitter’s world. Result. Math. 36 (1999) 57–68 · Zbl 0943.51013 · doi:10.1007/BF03322102
[14] Huang, W.: Adjacency preserving mappings of invariant subspaces of a null system. To appear Proc. Am. Math. Soc. · Zbl 0955.51004
[15] Klein, F.: Vorlesungen über Höhere Geometrie. 3. Aufl. Bearbeitet und herausgegeben von W. Blaschke. Grundlehren der math. Wissensch. in Einzeldarstellungen, Bd. 22, Springer-Verlag, Berlin, 1926 (Nachdruck 1968) · JFM 52.0624.09
[16] Leissner, W.: Büschelhomogene Lie-Ebenen. Journ. reine angew. Math. 246 (1971) 76–116
[17] Lie, S.: Über Komplexe, insbesondere Linien- und Kugelkomplexe, mit Anwendung auf die Theorie partieller Differentialgleichungen. Math. Ann. 5 (1872) 145–256 · JFM 04.0408.01 · doi:10.1007/BF01446331
[18] Lie, S. and Scheffers, G.: Geometrie der Berührungstransformationen. I. B.G. Teubner-Verlag, Leipzig, 1896
[19] Müller, E. and Krames, J.L.: Die Zyklographie. Leipzig, Wien, 1929
[20] Pinkall, U.: Dupin hypersurfaces. Math. Ann. 270 (1985) 427–440 · Zbl 0538.53004 · doi:10.1007/BF01473438
[21] Schröder, E.M.: Zur Kennzeichnung distanztreuer Abbildungen in nichteuklidischen Räumen. Journ. Geom. 15 (1980) 108–118 · Zbl 0463.51015 · doi:10.1007/BF01922487
[22] Study, E.: Vereinfachte Begründung von Lies Kugelgeometrie. I. Sitzungsber. der preuss. Akad. der Wissensch. 27 (1926) 360–380 · JFM 52.0630.02
[23] Takasu, T.: Differentialgeometrien in den Kugelräumen. II. Maruzen, Tokyo, 1939 · JFM 65.0783.02
[24] Takasu, T.: Parabolic Lie-Geometry. Yokohama Math. J. 4 (1956) 95–98 · Zbl 0074.15701
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.