Download 4-Dimensionale Translationsebenen mit genau einer by Betten D. PDF

By Betten D.

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The Extensibility of Local Lie Groups of Transformations and Groups on Surfaces', Ann. Math. 52 (1950), 606-636. 15. Pickert, G. :Projektive Ebenen, Springer, Berlin-G6ttingen-Heidelberg, 1955. 16. Plaumann, P. and Strambach, K. : 'ZweidimensionaleQuasik6rper mit Zentrum', Arch. Math. 21 (1970), 455-465. 17. Satzmann, H: 'Topological Planes', Adv. Math. 2 (1967), 1-60. 18. : 'Kompakte vier-dimensionale Ebenen', Arch. Math. 20 (1969), 551-555. 19. Salzmann, H. : 'Homomorphismen komplexer TerMrk6rper', Math.

Zurn Beweis der Stetigkeit von -c bemerken wir zun~ichst, dab die Beschr/inkung yon z auf das Innere und auf das AuBere der Parabel stetig ist. ~ ergibt sich sodann durch Grenztibergang s ~ - oo. Die Surjektivit~it v o n • folgt wie in [5 (d)]. An- 4 - D I M E N S I O N A L E TRA N S L A T I O N S E B E N E N 437 genommen, z w/ire linear, dann lieferte der A n s a t z f = A ~ + Bfl, 9 = Ca + Dfl mit Konstanten A, B, C und D, dab w = 0, z = 1, p = 0, q = - 1 ist. Dies w~ire ein Widerspruch zu den Existenzbedingungen.

D) Isomorphietypen und volle Kollineationsgruppe. F A L L 1. Die Gerade S bleibe unter der linearen Gruppe F o des Bfischels fest. Dann folgt wie in Satz 1 (c), dab Fo 3-dimensional ist. Sei (p eine lineare Abbildung des ~ auf sich, welche die Partition 23. . p,q in die Partition 23w,,~,,p,,q, fiberffihrt. Dann k6nnen wit wegen der Transitivit/it von A' auf R' annehmen, dab =(00 00) ist, dab also q~ die F o r m o) hat mit 2 x 2-reihigen Matrizen C und D. Bei Konjugation mit (p geht die Gruppe A in sich, und daraus folgt D = cC mit c # 0.

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