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In this article, we study THA-surfaces in the Galilean space 𝔾3, which has finite type immersion.

A Euclidean submanifold is said to be of Chen finite type if its coordinate functions are a finite sum of eigenfunctions of its Laplacian Δ. In this paper, we classify two types of affine homothetical surfaces of finite type in isotropic 3-space 핀 3 {\mathbb{I}^{3}} under the condition Δ ⁢ r i = λ i ⁢ r i {\Delta r_{i}=\lambda_{i}r_{i}} , where ( r i ) {(r_{i})} is the i-component function of the position vector r ( i = 1 , 2 , 3 {i=1,2,3} ), λ i ∈ ℝ {\lambda_{i}\in\mathbb{R}} and Δ denotes the Laplace operator. [ABSTRACT FROM AUTHOR]

A surface is minimal if the mean curvature ℋ mean {\mathcal{H}_{\rm mean}} vanishes everywhere. In this paper, we study some surfaces in the product space ℍ 2 × ℝ {\mathbb{H}^{2}\times\mathbb{R}} . In particular, we completely classify minimal surfaces. [ABSTRACT FROM AUTHOR]

In the three dimensional Lorentz-Heisenberg space Nil1³, a translation surface is parametrized by r(s; t) = γ1(s) γ2(t) or r(s; t) = γ2(t) * γ1(s); where γ1 and γ2 are curves contained in coordinate planes and * denotes the group operation of H3: In this paper we study translation surfaces in the 3-dimensional Heisenberg group H3 which is given by the Lorentz left invariant metrics whose mean curvature vanishes. [ABSTRACT FROM AUTHOR]

In this paper, we study triharmonic curves and bi -f-harmonic curves in the standard three-dimensional geometry Sol3 with the left-invariant metric gSol3 = ds² = (ezdx)² + (e-zdy)² + (dz)². We characterize the triharmonic curves in terms of their curvature and torsion. [ABSTRACT FROM AUTHOR]

In this paper we study translation surfaces with the nondegenerate third fundamental form in Lorentz-Minkowski space L3. As a result, we classify translation surfaces satisfying an equation in terms of the position vector field and the Laplace operator with respect to the third fundamental form III on the surface. [ABSTRACT FROM AUTHOR]

In this article, we study translation invariant surfaces in the 3-dimensional Heisenberg group H3 and the Lie group Sol3; which has FInite type immersion. [ABSTRACT FROM AUTHOR]

Following the publication online, it has been brought to the attention of the Editorial Board of Communications in Contemporary Mathematics (CCM) that the result of this paper was known since 1992 and appeared in the article by Luis Alías, Angel Ferrández and Pascual Lucas, Surfaces in the 3-dimensional Lorentz–Minkowski space satisfying Δx=Ax+B, Pacific J. Math. 156(2) (1992) 201–208.The information has been verified, and therefore the Editors-in-Chief of CCM have decided to retract the acceptance of this paper. [ABSTRACT FROM AUTHOR]

In this paper we study the helicoidal surfaces in the 3-dimensional Euclidean space under the condition ΔJr = Ar Ar; J = I, II, III, where A = (aij) is a constant 3 × 3 matrix and ΔJ denotes the Laplace operator with respect to the fundamental forms I, II and III. [ABSTRACT FROM AUTHOR]

In classical differential geometry, the problem of obtaining Gaussian and mean curvatures of a surface is one of the most important problems. In the Galilean and pseudo Galilean spaces, some special surfaces such as factorable surfaces, ruled Weingarten surfaces, translation surfaces and tubular surfaces have been studied in [2-4, 18, 19]. A regular surface M2 in Galilean space is called a area minimizing surface (minimal surface) if and only if its mean curvature is zero at each point. In this paper, we describe, THA-surfaces (Translation and homothetical affine surfaces) in the Galilean space G3 having constant Gaussian and mean curvatures and completely classify minimal or at THA-surfaces. Also, we classify linear Weingarten THA-surfaces in the Galilean space G3. [ABSTRACT FROM AUTHOR]

In this paper we study the translation surfaces in the 3-dimensional Euclidean and Lorentz-Minkowski space under the condition $${\Delta ^{III}r_{i} = \mu _{i}r_{i},\mu _{i} \in \mathbb{R}}$$, where Δ denotes the Laplacian of the surface with respect to the third fundamental form III. We show that in both spaces a translation surface satisfying the preceding relation is a surface of Scherk. [ABSTRACT FROM AUTHOR]

In this paper we classify the factorable surfaces in the three-dimensional Euclidean space $${\mathbb{E}^{3}}$$ and Lorentzian $${\mathbb{E}_{1}^{3}}$$ under the condition Δ r = λ r, where $${\lambda_{i}\in\mathbb{R}}$$ and Δ denotes the Laplace operator and we obtain the complete classification for those ones. [ABSTRACT FROM AUTHOR]