Abstract
We give a characterization of smooth stable and smooth projective planes in terms of submersion and transversality. Moreover, smooth affine translation planes are characterized by properties of their corresponding spreads, considered as subsets of the Grassmannian. The last section contains smoothness results about spherical Moebius planes. In particular, we establish smoothness properties of the classical Moebius plane.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bödi, R.: Smooth stable planes, Results. Math. 31 (1997), 300–321.
Bröcker, T. and Jönich, K.: Introduction to Differential Topology, Cambridge University Press, 1987.
Buekenhout, F. (Ed.): Handbook of Incidence Geometry, North-Holland, Amsterdam, 1995.
Greenberg, M. J.: Lectures on Algebraic Topology, Benjamin, Menlo Park (Cal.), 1967.
Grundhöfer, T.: Ternary fields of compact projective planes, Abh. Math. Sem. Univ. Hamburg 57 (1987), 87–101.
Grundhöfer, T. and Hähl, H.: Fibrations of spheres by great spheres over division algebras and their differentiability, J. Differential Geom. 31 (1990), 357–363.
Hähl, H.: Differentiable fibrations of the (2n —1)-sphere by great (n — 1)-spheres and their coordinatization over quasifields, Results Math. 12 (1987), 99–118.
Immervoll, S.: Glatte affine Ebenen und ihr projektiver Abschluβ, Diploma thesis, Tübingen 1998.
Löwen, R.: Vierdimensionale stabile Ebenen, Geom. Dedicata 5 (1976), 239–294.
Löwen, R.: Topology and dimension of stable planes: On a conjecture of H. Freudenthal, J. Reine Angew. Math. 343 (1983), 108–122.
Löwen, R.: Compact spreads and compact translation planes over locally compact fields, J. Geom. 36 (1989), 110–116.
Massey, W. S.: A Basic Course in Algebraic Topology, Springer, New York, 1991.
Otte, J.: Differenzierbare Ebenen, Dissertation, Kiel, 1992.
Otte, J.: Smooth projective translation planes, Geom. Dedicata 58 (1995), 203-212.
Salzmann, H., Betten, D., Grundhöfer, T., Hähl, H., Löwen, R. and Stroppel, M.: Compact Projective Planes, De Gruyter, Berlin, 1995.
Strambach, K.: Der Kreisraum einer sphärischenMöbiusebene, Monat. Math. 78 (1974), 156–163.
Wölk, R. D.: Topologische Möbiusebenen, Math. Z. 93 (1966), 311–333.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bödi, R., Immervoll, S. Implicit Characterizations of Smooth Incidence Geometries. Geometriae Dedicata 83, 63–76 (2000). https://doi.org/10.1023/A:1005273116586
Issue Date:
DOI: https://doi.org/10.1023/A:1005273116586