Abstract
We determine the class of all locally compact stable planesM of positive dimensiond≤4 which admit a reflection at each point of some open setU \(U \subseteq M\) M. Apart from the expected possibilities (planes defined by real and complex hermitian forms, and almost projective translation planes), one obtains (subplanes of)H. Salzmann's modified real hyperbolic planes [14; 5.3] and one exceptional plane which was not known before. The caseU=M has been treated [9] and is reproved here in a simpler way. The solution to the problem indicated in the title constitutes the main step in the proof of our results.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Betten, D.: Die komplex-hyperbolische Ebene. Math. Z.132, 249–259 (1973).
Dugundji, J.: Topology, 5th ed. Boston: Allyn & Bacon. 1970.
Groh, H.: Pasting ofR 2-planes. Preprint, Darmstadt 1977.
Löwen, R.: Vierdimensionale stabile Ebenen. Geom. Dedic.5, 239–294 (1976).
Löwen, R.: Halbeinfache Automorphismengruppen von vierdimensionalen stabilen Ebenen sind quasi-einfach. Math. Ann.236, 15–28 (1978).
Löwen, R.: Symmetric planes. Pacific J. Math.84, 367–390 (1979).
Löwen, R.: Classification of 4-dimensional symmetric planes. Math. Z.167, 137–159 (1979).
Löwen, R.: Central collineations and the parallel axiom in stable planes. Geom. Dedic. (To appear.)
Löwen, R.: Characterization of symmetric planes in dimension at most 4. Indag. Math. (To appear.)
Löwen, R.: Weakly flag homogeneous stable planes of low dimension. Arch. Math.33, 485–491 (1979).
Löwen, R.: Stable planes of low dimension admitting reflections at many lines. Resultate Math. (To appear.)
Montgomery, D., andL. Zippin: Periodic one-parameter groups in three-space. Trans. Amer. Math. Soc.40, 24–36 (1936).
Mostert, P. S.: On a compact group acting on a manifold. Ann. of Math.65, 447–455 (1957).
Salzmann, H.: Topological planes. Advanc. Math.2, 1–60 (1967).
Salzmann, H.: Kollineationsgruppen ebener Geometrien. Math. Z.99, 1–15 (1967).
Salzmann, H.: Kollineationsgruppen kompakter vierdimensionaler Ebenen. Math. Z.117, 112–124 (1970).
Salzmann, H.: Homogene 4-dimensionale affine Ebenen. Math. Ann.196, 320–322 (1972).
Strambach, K.: Zentrale und axiale Kollineationen in Salzmannebenen. Math. Ann.185, 173–190 (1970).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Löwen, R. Equivariant embeddings of low dimensional symmetric planes. Monatshefte für Mathematik 91, 19–37 (1981). https://doi.org/10.1007/BF01306955
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01306955