Skip to main content
SpringerLink
Log in

Pasting of R 2-planes

  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

The notion ‘pasting sum’ \(\underline{\underline {F\omega }}\) (P i , ℒ i ) of two R 2-planes (or Salzmann planes) (P i , ℒ i ) is developed. Necessary and sufficient conditions for it to be an R 2-plane again are given. The notion is applied to classify all flat projective planes whose collineation group contains a sub-group Δ with (isomorphism type of Δ, fixed element configuration)=(ℝ2, x).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Blbliography

  1. Betten, D., Ostmann, A.: ‘Wirkungen und Geometrien der Gruppe L 2 × R’. Geometriae Dedicata 7, 141–162 (1978).

    Google Scholar 

  2. Brouwer, L. E. J.: ‘Die Theorie der endlichen kontinuierlichen Gruppen, unabhängig von den Axiomen von Lie’. Math. Ann. 67, 246–267, (1909).

    Google Scholar 

  3. Dugundji, J.: Topology. 5th edition. Allyn and Bacon, Boston (1970).

    Google Scholar 

  4. Groh, H.: ‘R2-planes with 2-dimensional point transitive automorphism group’. Abh. Math. Sem. Hamburg 48, 171–202 (1979).

    Google Scholar 

  5. Groh H.: ‘Point homogeneous flat affine planes’. J. Geometry 8, 145–162, (1976).

    Google Scholar 

  6. Hewitt, E. Stromberg, K.: Real and Abstract Analysis. Springer, New York-Heidelberg-Berlin (1975).

    Google Scholar 

  7. Pickert, G.: Projektive Ebenen. Springer, New York-Heidelberg-Berlin (1975).

    Google Scholar 

  8. Salzmann, H.: ‘Kompakte zweidimensionale projektive Ebenen’. Math. Ann. 145, 401–428, (1962).

    Google Scholar 

  9. Salzmann, H.: ‘Zur Klassifikation topologischer Ebenen II’. Abh. Math. Sem. Hamburg 27, 145–166 (1964).

    Google Scholar 

  10. Salzmann, H.: ‘Topological planes’. Advances in Math. 2, 1–60, 1967.

    Google Scholar 

  11. Skornyakov, L. A.: ‘Curve systems in the plane’. Dokl. Akad. Nauk SSSR 98, 25–26 (1954).

    Google Scholar 

  12. Skornyakov, L. A.: ‘Curve systems in the plane’. Trudy Moskov. Mat. Obšč. 6, 135–164 (1957).

    Google Scholar 

  13. Strambach, K.: ‘Salzmannebenen mit hinreichend vielen Punkt- oder Geradenspiegelungen’. Math. Z. 99, 247–269 (1967).

    Google Scholar 

  14. Strambach, K.: ‘Zur Klassifikation von Salzmann-Ebenen mit 3-dimensionaler Kollineationsgruppe II’. Abh. Math. Sem. Hamburg 34, 159–169 (1970).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. FB 4 / AG 2 - Geometrie und Algebra -Technische Hochschule Darmstadt, Schloßgartenstraße 7, D-6100, Darmstadt, Bundesrepublik Deutschland

    Hansjoachim Groh

Authors
  1. Hansjoachim Groh
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Groh, H. Pasting of R 2-planes. Geom Dedicata 11, 69–98 (1981). https://doi.org/10.1007/BF00183191

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00183191

Keywords

Search

Navigation