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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.
53A20 Projective differential geometry
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