Abstract
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.
Highlights
A Euclidean submanifold is said to be of finite Chen type if its coordinate functions are finite sum of eigenfunctions of its Laplacian [1]
In [12, 13] Rahmani and Rahmani has showed that modulo an automorphism of the Lie algebra, the three-dimensional Lorentz Heisenberg group H3 has the following classes of leftinvariant Lorentz metrics: g1 1⁄4 Àdx2 þ dy2 þ ðxdy þ dzÞ2 g2 1⁄4 dx2 þ dy2 À ðxdy þ dzÞ2 g3 1⁄4 dx2 þ ðxdy þ dzÞ2 À 1⁄2ð1 À xÞdy À dz2: They proved that the metrics g1, g2, g3 are non-isometrics and that g3 is flat
Translation surfaces in ðH3; g3Þ In the Lorentz Heisenberg space H3, a translation surface is parametrized by rðx; yÞ 1⁄4 γ1ðxÞ * γ2ð yÞ, where γ1 and γ2 are two planar curves lying in planes, which are non-orthogonal and * denotes the group operation of H3
Summary
A Euclidean submanifold is said to be of finite Chen type if its coordinate functions are finite sum of eigenfunctions of its Laplacian [1]. Chen posed the problem of classifying the finite type surfaces in the three-dimensional Euclidean space E3. The notion of finite type can be extended to any function on a submanifold of a Euclidean space or a pseudo-. The authors would like to thank anonymous referees for their valuable comments and careful corrections to the original version of this paper
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