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Flat Laguerre planes of Kleinewillinghöfer type E obtained by cut and paste

Published online by Cambridge University Press:  17 April 2009

Günter F. Steinke
Günter F. Steinke
Affiliation:
Department of Mathematics and Statistics, College of Engineering University of Canterbury, Private Bag 4800, Christchurch 8020, New Zealand e-mail: G.Steinke@math.canterbury.ac.nz
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We provide examples of flat Laguerre planes of Kleinewillinghöfer type E, thus completing the classification of flat Laguerre planes with respect to Laguerre translations in B. Polster and G.F. Steinke, Results Maths. (2004). These planes are obtained by a method for constructing a new flat Laguerre plane from three given Laguerre planes devised in B. Polster and G. Steinke, Canad. Math. Bull. (1995) but no examples were given there.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 72 , Issue 2 , October 2005 , pp. 213 - 223
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Groh, H., ‘Topologische Laguerreebenen I.’, Abh. Math. Sem. Univ. Hamburg 32 (1968), 216–231.CrossRefGoogle Scholar
[2]Groh, H., ‘Topologische Laguerreebenen II’, Abh. Math. Sem. Univ. Hamburg 34 (1970), 11–21.CrossRefGoogle Scholar
[3]Kleinewillinghöfer, R., Eine Klassifikation der Laguerre-Ebenen, (Ph.D. Thesis) (TH, Darmstadt, 1979).Google Scholar
[4]Kleinewillinghöfer, R., ‘Eine Klassifikation der Laguerre-Ebenen nach ℒ-Streckungen und ℒ-Translationen’, Arch. Math. 34 (1980), 469–480.CrossRefGoogle Scholar
[5]Löwen, R. and Pfüller, U., ‘Two-dimensional Laguerre planes over convex functions’, Geom. Dedicata 23 (1987), 73–85.CrossRefGoogle Scholar
[6]Polster, B. and Steinke, G.F., ‘Cut and paste in 2-dimensional projective planes and circle planes’, Canad. Math. Bull. 38 (1995), 469–480.CrossRefGoogle Scholar
[7]Polster, B. and Steinke, G.F., ‘The inner and outer space of 2-dimensional Laguerre planes’, J. Austral. Math. Soc. Ser. A 62 (1997), 104–127.CrossRefGoogle Scholar
[8]Polster, B. and Steinke, G.F., ‘A family of 2-dimensional Laguerre planes of generalised shear type’, Bull. Austral. Math. Soc. 61 (2000), 69–83.CrossRefGoogle Scholar
[9]Polster, B. and Steinke, G.F., Geometries on Surfaces, Encyclopedia of Mathematics and its Applications (Cambridge University Press, Cambridge 84, 2001).CrossRefGoogle Scholar
[10]Polster, B. and Steinke, G.F., ‘On the Kleinewillinghöfer types of flat Laguerre planes’, Results Math. 46 (2004), 103–122.CrossRefGoogle Scholar
[11]Steinke, G.F., ‘Semiclassical topological flat Laguerre planes obtained by pasting along a circle’, Results Math. 12 (1987), 207–221.CrossRefGoogle Scholar
[12]Steinke, G.F., ‘Semiclassical topological flat Laguerre planes obtained by pasting along two parallel classes’, J. Geom. 32 (1988), 133–156.CrossRefGoogle Scholar