On the characterization of plane projective and complex MOEBIUS-transformations†
On the characterization of plane projective and complex MOEBIUS-transformations†
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DOI：
10.1002/mana.19670330506
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发表时间：
1967
影响因子：
1
通讯作者：
J. Aczél;M. Mckiernan
中科院分区：
文献类型：
--
作者：
J. Aczél;M. Mckiernan
MoEBIus-transformations (and their Conjugates) have been characterized as the only circle preserving transformations of the completed complex plane into itself (eg in [5, 3, 4, 61). Similarly the plane projective transformations were characterized as the only straight line preserving transformations of the real projective plane into itself, eg in [5, 2, 71. These characterizations involve certain suppositions, as eg in [S] that the transformations mapped all circles onto circles and that the mappings were oneto-one (schlicht) everywhere.After recapitulating some known facts on complex functions and on functional equations in section 1, we give in section 2 of this paper a slightly simplified variant of the argument of [6]. In section 3 we show by another argument, that in order to characterize the MoEBIus-transformations (and their conjugates), it is enough to assume that the mapping is one-to-one in four points and to consider (in the suppositions as well as in the result) only points of three circles. In section 4 we get a similar result which gives a generalization of a fundamental theorem of projective geometry, see eg 171-a characterization of the projective transformations of points of four straight lines in the real plane, assuming only that collinear points of these four straight lines are mapped into collinear points and that the mapping is one-to-one in the points of intersection of these four straight lines. Finally, we show in section 5, that our suppositions were in a certain sense minimal: there exist also other one-to-one mappings carrying collinear points of three straight lines into collinear points and we determine all of them. This will be done by finding the general real solution of the somewhat unusual functional equation
