Invariant surfaces with coordinate finite-type Gauss map in simply isotropic space. (English) Zbl 1468.53006
A surface is said to have coordinate finite-type Gauss map if \(\Delta G=-AG\), where \(G\) is the Gauss map of the surface and \(A=\operatorname{diag}(\lambda_1,\lambda_2,\lambda_3)\) is a diagonal matrix. A surface in the simply isotropic space \(\mathbb{I}^3\) is said to be invariant if there exists a 1-parameter subgroup that leaves it invariant. The aim of this paper is to characterize the invariant surfaces with coordinate finite-type Gauss map in \(\mathbb{I}^3\).
The authors consider two different normal Gauss maps: The first one, \(\mathbf{N}_m\), is called the minimal normal, since the trace of \(d\mathbf{N}_m\) vanishes identically, while the second one, \(G\), is called the parabolic normal, since it takes values on a unit sphere of parabolic type. In both cases, the authors obtain explicit formulas for the invariant coordinate finite-type Gauss map surfaces. Surfaces with harmonic normals, minimal or parabolic, without the invariant assumption, are also studied in the paper.
The authors consider two different normal Gauss maps: The first one, \(\mathbf{N}_m\), is called the minimal normal, since the trace of \(d\mathbf{N}_m\) vanishes identically, while the second one, \(G\), is called the parabolic normal, since it takes values on a unit sphere of parabolic type. In both cases, the authors obtain explicit formulas for the invariant coordinate finite-type Gauss map surfaces. Surfaces with harmonic normals, minimal or parabolic, without the invariant assumption, are also studied in the paper.
Reviewer: Marcos Craizer (Rio de Janeiro)
Keywords:
simply isotropic space; Gauss map; helicoidal surface; parabolic revolution surface; invariant surface; Cayley-Klein geometryReferences:
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