Metric structures imposed on Grassmannians

Abstract

In the note we show that the conjugacy of points determined by a correlation of a space of pencils is definitionally equivalent to (suitably defined) orthogonality of lines and to a ‘polarity’, introduced in the space in question. This proves that ‘metric geometry of spaces of pencils’ can be equivalently developed as a theory of incidence enlarged by a natural (point \(\leftrightarrow \) line) or (line \(\leftrightarrow \) line) relation.

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References

  1. Bichara, A., Tallini, G.: On a characterization of Grassmann space representing the h-dimensional subspaces in a projective space. Ann. Discrete Math. 18, 113–132 (1983)

    MathSciNet  MATH  Google Scholar 

  2. Cameron, P.J.: Projective and polar spaces, QMW Math Notes 13. University of London, Queen Mary and Westfield College (1992)

    Google Scholar 

  3. Cohen, M.A.: Point-line spaces related to buildings. In: Buekenhout, F. (ed.) Handbook of incidence geometry, pp. 647–737. North-Holland, Amsterdam (1995)

    Google Scholar 

  4. Coxeter, H.S.M.: Projective geometry. Springer, Berlin (2003)

    Google Scholar 

  5. Faure, C.-A., Frölicher, A.: Dualities for infinite-dimensional projective geometries. Geom. Dedicata 56, 225–236 (1995)

    MathSciNet  Article  Google Scholar 

  6. Faure, C.-A., Frölicher, A.: Modern projective geometry. In: Hazewinkel, M. (ed.) Mathematics and its Applications, vol. 521. Kluwer Academic Publishers, Dordrecht (2000)

    Google Scholar 

  7. Giering, O.: Vorlesungen über höhere Geometrie. Vieweg, Braunschweig Wiesbaden (1982)

    Google Scholar 

  8. Kordos, M.: Foundations of projective and metric projective geometry (in Polish). PSP, Warsaw (1984)

    Google Scholar 

  9. Onishchik, A.L., Sulanke, R.: Projective and Cayley-Klein Geometries. Springer-Verlag, Berlin (2006)

    Google Scholar 

  10. Prażmowski, K., Żynel, M.: Orthogonality of subspaces in metric-projective geometry. Adv. Geom. 11(1), 103–116 (2011)

    MathSciNet  Article  Google Scholar 

  11. Schröder, E.M.: Metric geometry. In: Buekenhout, F. (ed.) Handbook of incidence geometry, pp. 945–1013. North-Holland, Amsterdam (1995)

    Google Scholar 

  12. Tallini, G.: Partial line spaces and algebraic varieties. Sympos. Math. 28, 203–217 (1986)

    MathSciNet  MATH  Google Scholar 

  13. Żynel, M.: Correlations of spaces of pencils. J. Appl. Logic 10(2), 187–198 (2012)

    MathSciNet  Article  Google Scholar 

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Correspondence to Mariusz Żynel.

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Prażmowski, K., Żynel, M. Metric structures imposed on Grassmannians. Beitr Algebra Geom 61, 507–513 (2020). https://doi.org/10.1007/s13366-019-00480-9

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Keywords

  • Polarity
  • Conjugacy
  • Space of pencils
  • Orthogonality
  • Metric projective geometry

Mathematics Subject Classification

  • 51F20
  • 51A50 (14M15)