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On special plane nets. (English) Zbl 0763.53015
By a complex parameter transformation the hyperbolic Laplace equation of a conjugate net carries over to an elliptic equation which can be written as $$x_{z\overline z}=ax_ z+\overline ax_{\overline z}+cx$$ (using complex numbers $$z$$ in the parameter plane). Nets of this kind in $$\mathbb{P}^ 3(\mathbb{R})$$ are called “elliptic nets”. Applying the well-known theory of conjugate nets, the notions of Laplace-transformations and Darboux invariants (which are written as 2-forms rather than functions) can be defined for elliptic nets. Such a net is called “special” if $$x_ 2$$ does not exist or is situated on the line $$x_ 1, x_{-1}$$ (indices indicating the Laplace transforms). For special elliptic nets, the same condition is true for $$x_{-2}$$ instead of $$x_ 2$$. Two standard examples of special elliptic nets are given; they are induced by the Riemannian metric on the sphere and the hyperbolic metric of the Cayley-Klein model and are denoted by $$N^ +$$ and $$N^ -$$ respectively.
The following is one of the very rare global results in projective differential geometry: $$N^ +$$ is characterized by the following properties: (i) $$N$$ is a special elliptic net defined in $$\mathbb{C}$$, (ii) the Darboux forms $$\varphi_ 1,\varphi_{-1}$$ are equal and (say positive) definite, (iii) the corresponding Gauss curvature is positive everywhere.
MSC:
 53A20 Projective differential geometry
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References:
 [1] G. Fubini E. Čech: Introduction à la géométrie projective différentielle des surfaces. Gauthier-Villars, Paris, 1931. · JFM 57.0936.01 [2] E. P. Lane: Projective differential geometry of curves and surfaces. The University of Chicago Press, 1932. · Zbl 0005.02501 [3] G. Tzitzéica: Géométrie différentielle projective des réseaux. Académie roumanei, Bucarest, Bucarest, 1923. · JFM 57.0952.04
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