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Differentialgeometrie des isotropen Raumes. IV: Theorie der flächentreuen Abbildungen der Ebene. (German) Zbl 0063.07216
[Parts I, II and III see Sitzungsber. Akad. Wiss. Wien, Math.-Naturw. Kl., Abt. IIa 150, 1–53 (1941; Zbl 0026.08102); Math. Z. 47, 743–777 (1942; Zbl 0026.26301); 48, 369–427 (1942; Zbl 0027.25301).]
Equiareal transformations of the sphere have been investigated by means of their close relationship to the theory of surfaces in elliptic space [see, for example, G. Fubini, Ann. Scuola Norm. Super. Pisa 9, No. 1 (1904; JFM 35.0668.03) or L. Bianchi, Lezioni di Geometria Differenziale, 3rd ed., v. 2, Pisa (1923; JFM 49.0498.06)]. This paper uses a similar method for the investigation of equiareal transformations of the plane by means of their relationship to the theory of surfaces in isotropic three-dimensional space. The underlying principle is the possibility of mapping the plane elements of a surface (more generally, a two-dimensional set of plane elements in the sense of Lie) on the point pairs of a plane by means of the two Clifford parallelisms. Such a “paratactic” mapping is equiareal, and every equiareal mapping can be considered as a “paratactic” mapping. The author proceeds to a classification of equiareal transformations and discusses many special cases. Curves in isotropic space correspond to transformations of which the locus of the middle points is one-dimensional; other interesting transformations correspond to minimal surfaces or to surfaces of constant relative curvature \(K=+1\). These last mappings have a close relation to the theory of complex analytic curves. There is also a very readable summary of the differential geometry of curves and surfaces in isotropic space.

MSC:
53Axx Classical differential geometry
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