Translation surfaces in Euclidean space with constant Gaussian curvature

Hasanis, Thomas; López, Rafael

doi:10.48550/arxiv.1809.02758

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2024

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Cited by 2 publications

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“…Indeed, the translation surface x given by x(u, v) = u, v, 1 a log cos(av) cos(au) , a > 0 is a minimal surface (the surface is called the Scherk surface). Recently, classifications of translation surfaces in the Euclidean 3-space under some conditions on its curvatures are studied (for instance, [10,[19][20][21]). Moreover, translation surfaces in the Lorentz-Minkowski space are applied to the study of singular surfaces (for instance, [1]).…”

Section: Introductionmentioning

confidence: 99%

Singularities of Translation Surfaces in the Euclidean 3-Space

Fukunaga

Takahashi

2022

Results Math

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Section: Introductionmentioning

confidence: 99%

Singularities of Translation Surfaces in the Euclidean 3-Space

Fukunaga

Takahashi

2022

Results Math

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Some generic hypersurfaces in a Euclidean space

Alohali,

Deshmukh

2024

MATH

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<abstract><p>In this paper, we find three nontrivial characterizations of Euclidean spheres. In the first result, we show that the existence of a nonzero nontrivial concircular vector field $ {\omega } $ on a compact and connected hypersurface $ N $ of the Euclidean space $ R^{m+1} $ with a mean curvature $ \alpha $ constant along the integral curves of $ {\omega } $ and a shape operator $ T $ satisfying $ T({\omega) = \alpha \omega } $ implies that $ \alpha $ is a constant and $ N $ is isometric to a sphere, and the converse also holds. In the second result, we show that the presence of a unit Killing vector field $ \mathbf{v} $ on a compact and connected hypersurface $ N $ of a Euclidean space $ R^{m+1} $ gives a nonzero function $ \sigma = g\left(T \mathbf{v}, \mathbf{v}\right) $ with shape operator $ T $, and the integral of the function $ m\alpha \sigma Ric\left(\mathbf{v}, \mathbf{v}\right) $ has a certain lower bound, and is isometric to an odd-dimensional sphere, and the converse holds too. Finally, we show that for a compact and connected hypersurface $ N $ with support $ \rho $ and basic vector field $ \mathbf{u} $, the integral of the Ricci curvature $ Ric\left(\mathbf{u}, \mathbf{u}\right) $ has a specific lower bound and is necessarily isometric to a sphere, and the converse also holds.</p></abstract>

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Translation surfaces in Euclidean space with constant Gaussian curvature

Abstract: We prove that the only surfaces in 3-dimensional Euclidean space R 3 with constant Gaussian curvature K and constructed by the sum of two space curves are cylindrical surfaces, in particular, K = 0.

Cited by 2 publications

References 9 publications

Singularities of Translation Surfaces in the Euclidean 3-Space

Singularities of Translation Surfaces in the Euclidean 3-Space

Some generic hypersurfaces in a Euclidean space

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