Complex Curves as Lines of Geometries

Autor: Olga Belova, Karl Strambach, Josef Mikeš
Rok vydání: 2016
Předmět:
Zdroj: Results in Mathematics. 71:145-165
ISSN: 1420-9012
1422-6383
Popis: We investigate Hjelmslev geometries \({\mathcal{H}}\) having a representation in a complex affine space \({\mathbb{C}^n}\) the lines of which are given by entire functions. If \({\mathcal{H}}\) has dimension 2 and the entire functions satisfy some injectivity conditions, then \({\mathcal{H}}\) is a substructure of the complex Laguerre plane. If the lines are geodesics with respect to a natural connection \({\nabla^{\circ}}\), then a detailed classification of them as well as of the corresponding geometries is obtained. Generalizations of complex Grunwald planes play a main role in the classification. Since in the considered geometries the set of lines is invariant under the translation group of \({\mathbb{C}^n}\), we classify all complex curves C in \({\mathbb{C}^n}\) given by entire functions as well as the connections \({\nabla^\circ}\) such that all images of C under the translation group of \({\mathbb C^n}\) consist of geodesics with respect to \({\nabla^{\circ}}\).
Databáze: OpenAIRE
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