Download e-book for iPad: Aufbau der Geometrie aus dem Spiegelungsbegriff by Friedrich Bachmann

By Friedrich Bachmann

Torischen Gruppenelemente sind und in den en wir geometrische Bezie hungen wie Inzidenz undOrthogonalitat durch gruppentheoretische Rela tionen erklaren. Die rein gruppentheoretisch formulierten Axiome, die wir wahlen, stellen einfache geometrische Aussagen flir die Punkte und Geraden der metrischen Ebenen dar. Dementsprechend kann guy beim Beweisen aus den Axiomen die Vorteile des gruppentheoretischen Kalktils ausnutzen, ohne den Leitfaden der Anschauung aus der Hand zu geben. Bemerkenswert ist, wie wenige Axiome notig sind. Die metrischen Ebenen, die mit den axiomatisch gegebenen Gruppen definiert sind, sind daher von recht allgemeiner Natur. Eine metrische Ebene braucht nicht anordenbar (erst recht nicht stetig) zu sein. In einer metrischen Ebene braucht nicht freie Beweglichkeit zu bestehen. Es gibt auch metrische Ebenen mit nur endlich vielen Punkten und Geraden. Der Begriff der metrischen Ebene enthalt keine Entscheidung tiber die Parallelenfrage, d.h. tiber die Frage nach dem Schneiden oder Nicht schneiden der Geraden. Die ebene metrische Geometrie, die wir ent wickeln, enthalt ebene euklidische, hyperbolische und elliptische Geo metrie als Spezialfalle, und wird daher, mit einem Ausdruck von J. BOLYAI, auch ebene absolute Geometrie genannt.

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First, we assume that 111 is not empty. Then 111 is a non empty open part of M and in each point p of Ill, we know that (Vh)p = 0. The classical Pick-Berwald theorem then implies that 111 is an open part of a nondegenerate ellipsoid or hyperboloid. Thus detS is a constant different from zero on 111" The continuity of detS then implies that fit = M. Finally, we may assume that S = 0 on the whole of M. Thus by Proposition 2, we can suppose that M is given by the equation z -- P(x,y), where P is a polynomial of degree at most k + 1, and that the canonical affine normal vector field is given by (0,0,1).

N be the Frenet frame of its directrix curve ~o" through ~p2 if and only if (i) n = 2m, (ii) ~ = ~m, (iii) ~o(z) Theorem 5 where z is a stereographic complex coordinate. - ~n(-I/~) be a linearly full minimal L e t %b : S 2 --+ CP n, n - 2m, immersion which factors through ~P~, ~o is k-point ramified for k < 2. and suppose that the directrix curve Let z be a stereographic complex coordinate such that if k ~ 0 then 4o has a higher order singularity at z - O. Then ~o is given by g/o(Z) = /kp zk' + ' " "+ kp ep (t) p-O where e_o .

Remark The condition of factoring through RP 2 is crucial here. result is not true in general. [l]). This, together with Theorem 2, is a first step in dealing with the space of minimal immersions of S ~ into Sn. Finally, combining Theorems 2 and 6, we have the following partial generalisation of Theorem 5. Theorem 7 Let # : S ~ --+ CP n be a linearly full minimal immersion which factors through Rp2. Then up to holomorphic isometries of CP n there are only finitely many minimal immersions of S 2 into CP n which factor through ~p2 and have the same singularity type as ~.

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Aufbau der Geometrie aus dem Spiegelungsbegriff by Friedrich Bachmann


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