Invariant surfaces with coordinate finitetype Gauss map in simply isotropic space
Abstract
We consider the extrinsic geometry of surfaces in simply isotropic space, a threedimensional space equipped with a rank 2 metric of index zero. Since the metric is degenerate, a surface normal cannot be unequivocally defined based on metric properties only. To understand the contrast between distinct choices of an isotropic Gauss map, here we study surfaces with a Gauss map whose coordinates are eigenfunctions of the surface LaplaceBeltrami operator. We take into account two choices, the socalled minimal and parabolic normals, and show that when applied to simply isotropic invariant surfaces the condition that the coordinates of the corresponding Gauss map are eigenfunctions leads to planes, certain cylinders, or surfaces with constant isotropic mean curvature. Finally, we also investigate (nonnecessarily invariant) surfaces with harmonic Gauss map and show this characterizes constant mean curvature surfaces.
 Publication:

arXiv eprints
 Pub Date:
 April 2020
 arXiv:
 arXiv:2004.12221
 Bibcode:
 2020arXiv200412221K
 Keywords:

 Mathematics  Differential Geometry;
 53A35;
 53B25;
 53C42
 EPrint:
 26 pages, 3 figures