Complex Curves as Lines of Geometries

Abstract

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 Grünwald 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 ${\nabla^{\circ}}$.