Helicoidal surfaces in the three-dimensional Lorentz–Minkowski space satisfying Δ II r i  = λ i r i

PurposeIn the Lorentz Heisenberg space H3 endowed with flat metric g3, a translation surface is parametrized by r(x, y) = γ1(x)*γ2(y), where γ1 and γ2 are two planar curves lying in planes, which are not orthogonal. In this article, we classify translation surfaces in H3, which satisfy some algebraic equations in terms of the coordinate functions and the Laplacian operator with respect to the first fundamental form of the surface.Design/methodology/approachIn this paper, we classify some type of space-like translation surfaces of H3 endowed with flat metric g3 under the conditionΔri = λiri. We will develop the system which describes surfaces of type finite in H3. For solve the system thus obtained, we will use the calculation variational. Finally, we will try to give performances geometric surfaces that meet the condition imposed.FindingsClassification of six types of translation surfaces of finite type in the three-dimensional Lorentz Heisenberg group H3.Originality/valueThe subject of this paper lies at the border of geometry differential and spectral analysis on manifolds. Historically, the first research on the study of sub-finite type varieties began around the 1970 by B.Y.Chen. The idea was to find a better estimate of the mean total curvature of a compact subvariety of a Euclidean space. In fact, the notion of finite type subvariety is a natural extension of the notion of a minimal subvariety or surface, a notion directly linked to the calculation of variations. The goal of this work is the classification of surfaces in H3, in other words the surfaces which satisfy the condition/Delta (ri) = /Lambda (ri), such that the Laplacian is associated with the first, fundamental form.

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“…In [5], Baba-Hamed, Bekkar and Zoubir studied coordinate finite type translation surface in a three-dimensional Minkowski Space.…”

Section: Introductionmentioning

confidence: 99%

Classification of space-like translation surfaces in the 3-dimensional Lorentz Heisenberg group H3

Medjati

Hanifi

Medjahdi

2021

AJMS

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“…It is well known that a helicoidal surface is a generalization of a rotation surface. There are many studies about these surfaces under some given certain conditions [1][2][3][4][5][6][7][8][9][10][11][12]. Recently, the popular question has become whether a helicoidal surface can be constructed when its curvatures are prescribed.…”

Section: Introductionmentioning

confidence: 99%

Constructions of Helicoidal Surfaces in a 3-Dimensional Complete Manifold with Density

Yıldız¹

2018

Mathematics

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“…In [1], the named author with M. Bekkar studied the helicoidal surfaces without parabolic points in the three-dimensional Lorentz-Minkowski space R 3 1 which satisfy…”

Section: Introductionmentioning

confidence: 99%

Helicoidal surfaces satisfying $${\Delta ^{II}\mathbf{G}=f(\mathbf{G}+C)}$$ Δ II G = f ( G + C )

Baba-Hamed

2015

J. Geom.

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In this paper, we study helicoidal surfaces without parabolic points in Euclidean 3-space R 3 , satisfying the condition Δ II G = f (G + C), where Δ II is the Laplace operator with respect to the second fundamental form, f is a smooth function on the surface and C is a constant vector. Our main results state that helicoidal surfaces without parabolic points in R 3 which satisfy the condition Δ II G = f (G + C), coincide with helicoidal surfaces with non-zero constant Gaussian curvature.Mathematics Subject Classification. Primary 53A05; Secondary 53B25, 53C40.

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Helicoidal surfaces in the three-dimensional Lorentz–Minkowski space satisfying Δ II r i = λ i r i

Cited by 6 publications

References 5 publications

Classification of space-like translation surfaces in the 3-dimensional Lorentz Heisenberg group H3

Classification of space-like translation surfaces in the 3-dimensional Lorentz Heisenberg group H3

Constructions of Helicoidal Surfaces in a 3-Dimensional Complete Manifold with Density

Helicoidal surfaces satisfying $${\Delta ^{II}\mathbf{G}=f(\mathbf{G}+C)}$$ Δ II G = f ( G + C )

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