Invariant surfaces with coordinate finite-type Gauss map in simply isotropic space

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:495

Article

Kelleci, Alev¹  da Silva, Luiz C. B.² 

[1] Firat Univ, Dept Math, TR-23200 Elazig, Turkey

[2] Weizmann Inst Sci, Dept Phys Complex Syst, IL-7610001 Rehovot, Israel

关键词: Simply isotropic space;    Gauss map;    Helicoidal surface;    Parabolic revolution surface;    Invariant surface;    Cayley-Klein geometry;   

DOI  :  10.1016/j.jmaa.2020.124673

来源: Elsevier

【 摘 要 】

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 Laplace-Beltrami operator. We take into account two choices, the so-called 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 (non necessarily invariant) surfaces with harmonic Gauss map and show this characterizes constant mean curvature surfaces. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free