Abstract
A. Cayley [5] and F. Klein [12] were the first mathematicians who introduced the notion of a metric in a real projective space, by specializing a set of quadrics (called the absolute). We show in this paper, that from an algebraic point of view these projective spaces can be described by vector spaces, in which a metric structure is given by one or more symmetric bilinear forms. We study these Cayley- Klein vector spaces and their groups of automorphisms (orthogonal Cayley- Klein groups) over arbitrary commutative fields of characteristic ≠ 2.
Similar content being viewed by others
References
Artin, E., Geometric Algebra. Interscience, New York 1957.
Bachmann, F., Aufbau der Geometrie aus dem Spiegelungsbegriff. Second supplemented edition. Springer, Heidelberg-Berlin, 1973.
Birkhoff, G., Lattice Theory, Third edition, American Math. Soc. Colloquium Publ. Vol. 25, Revised Edition 1973.
Cayley, A., A sixth memoir upon quantics. Phil. Trans. R. Soc, London, 1859 - cp. Collected Math. Papers, Vol. 2. Cambridge 1889.
Coxeter, H.S.M., The Real Projective Plane, Second edition, Cambridge University, Press, London 1955.
Dieudonnè, J., La gèomètrie des groupes classiques. Berlin 1955.
Faure, C.-A. and A. Frölicher, Modem Projective Geometry. Kluwer, Dordrecht 2000.
Giering, O., Vorlesungen über höhere Geometrie. Vieweg, Braunschweig 1982.
Grove, L.C., Classical Groups and Geometric Algebra, AMS, Rhode Island 2002.
Huppert, B., Geometric Algebra, U. of Illinois Chicago Lecture Notes, 1968/1969.
Karzel, H., Geschichte der Geometrie seit Hilbert. Wissenschaftliche Buchgesellschaft, Darmstadt 1988.
Klein, F., Über die sogenannte Nicht-euklidische Geometrie. Math. Ann. Vol. 4 (1871), 573–625 and Vol. 6 (1873), 112-145.
Klein, F., Vorlesungen über nicht-euklidische Geometrie. Springer, Berlin 1928.
Lenz, H., Vorlesungen über projektive Geometrie. Akademische Verlagsgesellschaft, Leipzig 1965.
Lingenberg, R., Metric Planes and Metric Vector Spaces. John Wiley, New York 1979.
O’Meara, O. T., Introduction to Quadratic Forms. Springer, Berlin 1963.
Rosenfeld, B.A., Non-euclidean Spaces, Moscow, Nauka 1969 (russian).
Snapper, E. and Troyer, J., Metric Affine Geometry. New York 1971.
Sommerville, D. M. Y., Classification of geometries with projective metrics. Proc. Edinburgh. Math. Soc. 28 (1910-1911), 25–41.
Struve, H. and R. Struve, Eine synthetische Charakterisierung der Cayley-Kleinschen Geometrien. Zeitschr. f. Math. Logik und Grundlagen d. Math. 31 (1985), 569–573.
Struve, H. and R. Struve, Endliche Cayley-Kleinsche Geometrien. Arch. Math. 48 (1987), 178–184.
Struve, H. and R. Struve, Zum Begriff der projektiv-metrischen Ebene. Zeitschr. f. Math. Logik und Grundlagen d. Math. 34 (1988), 79–88.
Struve, H. and R. Struve, Projective spaces with Cayley-Klein metrics. J. of Geo. 81 (2004), 155–167.
Struve, H. and R. Struve, Klassische nicht-euklidische Geometrien - ihre historische Entwicklung und Bedeutung und ihre Darstellung: Teil I and Teil II. Math. Semesterber. 51 (2004), 37–67 and 207-223.
Taylor, D.E., The Geometry of the Classical Groups, Heldermann Verlag, Berlin 1992.
Veblen, O. and J. W. Young, Projective geometry, vol 1,2. Boston 1910, 1918.
Wolff, H. and J. Ahrens and A. Dreß, Relationen zwischen Symmetrien in orthogonalen Gruppen. J. reine angew. Math. 234, 1–11(1969).
Yaglom, I. M., A Simple Non-Euclidean Geometry and Its Physical Basis. Springer, Berlin 1979.
Yaglom, I. M. and B. A. Rosenfeld and E. U. Yasinskaya, Projective Metrics. Russian mathematical surveys Vol 19 (1964), 49–107.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Struve, R. Orthogonal Cayley-Klein Groups. Results. Math. 48, 168–183 (2005). https://doi.org/10.1007/BF03322905
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03322905