Abstract
Affine polar spaces are polar spaces from which a hyperplane (that is a proper subspace meeting every line of the space) has been removed. These spaces are of interest as they constitute quite natural examples of ‘locally polar spaces’. A characterization of affine polar spaces (of rank at least 3) is given as locally polar spaces whose planes are affine. Moreover, the affine polar spaces are fully classified in the sense that all hyperplanes of the fully classified polar spaces (of rank at least 3) are determined.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bourbaki, N., Groupes et algèbres de Lie, Chapitres IV, V, VI, Hermann, Paris, 1968.
Buekenhout, F. and Sprague, A., ‘On the foundations of polar geometry’, Geom. Dedicata 3 (1974).
Buekenhout, F. and Sprague, A. ‘Polar spaces having some line of cardinality two’, J. Combin. Theory Ser. A 33 (1982), 223–228.
Hall, J. I. and Shult, E. E. ‘Locally cotriangular graphs’, Geom. Dedicata 18 (1985), 113–159.
Johnson, P. and Shult, E. E., ‘Local characterizations of polar spaces’, Geom. Dedicata 28 (1988), 127–151.
Teirlinck, L., ‘On linear spaces in which every plane is either projective or affine’, Geom. Dedicata 4 (1975), 39–44.
Tits, J., ‘Buildings of spherical type and finite BN-pairs’, Lecture Notes in Math. 386, Springer, Berlin, 1974.
Tits, J. L., ‘Classification of algebraic semisimple groups’, Algebraic Groups and Discontinuous Subgroups, Boulder 1965, Proceedings of Symp. in Pure Math., Vol IX, Amer. Math. Soc., Providence, 1966, pp. 33–62.
Veldkamp, F. D., ‘Polar geometry I–IV’, Indag. Math. 21 (1959), 512–551.
Author information
Authors and Affiliations
Additional information
In honor of J. Tits on the occasion of his sixtieth birthday
E.E.S. was partially supported by the National Science Foundation, U.S.A.
Rights and permissions
About this article
Cite this article
Cohen, A.M., Shult, E.E. Affine polar spaces. Geom Dedicata 35, 43–76 (1990). https://doi.org/10.1007/BF00147339
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00147339