Weighted minimal affine translation surfaces in Euclidean space with density

Yoon, Dae Won; Yüzbaşı, Zühal Küçükarslan

doi:10.1142/s0219887818501967

Cited by 11 publications

(8 citation statements)

References 9 publications

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“…Also, Belarbi and et al have studied the surfaces in 3 with density and they have given some results in a Riemannian manifold M with density in [16] and [26]. Next, ruled minimal surfaces in 3 with density ; helicoidal surfaces in 3 with density − 2 − 2 and weighted minimal affine translation surfaces in Euclidean space with density have been studied in [27,23,24], respectively. Also, some types of surfaces have been studied by geometers in other spaces such as Minkowski 3-space and Galilean 3-space with density.…”

Section: Introductionmentioning

confidence: 99%

Rotational Hypersurfaces in Euclidean 4-Space with Density

Altin

2022

Politeknik Dergisi

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Section: Introductionmentioning

confidence: 99%

Rotational Hypersurfaces in Euclidean 4-Space with Density

Altin

2022

Politeknik Dergisi

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“…It is said that, a hypersurface is weighted minimal and weighted flat if the weighted mean curvature and weighted Gaussian curvature vanish, respectively. After defining these notions, lots of studies have been done in different spaces with different densities by giving important characterizations for some types of curves and surfaces (for instance, [1][2][3][4]6,[12][13][14]16,17,19] and etc).…”

Section: Introductionmentioning

confidence: 99%

Yoğunluklu Öklidyen 3-Uzayında Helis Yüzeyleri

Kazan¹,

Kazan²

2021

European Journal of Science and Technology

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The differential geometry of helix curves and helix hypersurfaces in different spaces has important application areas in many disciplines. Also, the notion of weighted manifold is become to be a very popular topic for scientists in recent years. In this context, after defining the notions of weighted mean curvature (or -mean curvature) and weighted Gaussian curvature (or -Gaussian curvature) of an n-dimensional hypersurface on manifolds with density, lots of studies have been done by differential geometers in different spaces with different densities. So, in the present study, firstly we give the normal vector field, mean curvature and Gaussian curvature of a helix surface in three dimensional Euclidean space and after that, we obtain the weighted mean curvature and weighted Gaussian curvature of a helix surface generated by a unit speed planar curve in three dimensional Euclidean space with different three densities by stating the parametric equation of this surface. However, we know that a hypersurface is weighted minimal and weighted flat in Eucilidean 3-space with density if the weighted mean curvature and the weighted Gaussian curvature vanish, respectively. So, by using these definitions, we obtain the weighted minimal helix surfaces for these different densities and give some results for weighted flatness of the helix surfaces in Euclidean 3-space. We hope that, this study will bring a new viewpoint to differential geometers who are dealing with constant angle surfaces and in near future, one can handle these surfaces in different spaces with another densities.

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“…A hypersurface is called weighted flat (or -flat), if its weighted Gaussian curvature vanishes. After these definitions, lots of studies have been done by differential geometers about weighted manifolds, for instance [16][17][18][19][20][21][22][23][24][25]. Let we take ( ) = (ℎ ) × ( ) −1 .…”

Section: Introductionmentioning

confidence: 99%

Monge Hypersurfaces in Euclidean 4-Space with Density

2020

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Bu çalışmada, ilk olarak 4-boyutlu Öklidyen uzayında bir Monge hiperyüzeyinin ortalama ve Gaussian eğriliklerini verdik. Ardından, farklı yoğunluklara sahip uzayında Monge hiperyüzeylerini çalıştık. Bu bağlamda, ve hepsi aynı anda sıfır olmayan sabitler olmak üzere, (lineer yoğunluk) ve yoğunluklu uzayında ağırlıklı minimal ve ağırlıklı flat Monge hiperyüzeylerini ve sabitlerinin farklı seçimleri yardımıyla elde ettik. Anahtar Kelimeler:Yoğunluklu manifold, ağırlıklı ortalama eğrilik, ağırlıklı gaussian eğriliği, monge yüzeyleri.

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Weighted minimal affine translation surfaces in Euclidean space with density

Abstract: The aim of this work is to study affine translation surfaces in the Euclidean 3-space with density. We completely classify affine translation surfaces with zero weighted mean curvature.

Cited by 11 publications

References 9 publications

Rotational Hypersurfaces in Euclidean 4-Space with Density

Rotational Hypersurfaces in Euclidean 4-Space with Density

Yoğunluklu Öklidyen 3-Uzayında Helis Yüzeyleri

Monge Hypersurfaces in Euclidean 4-Space with Density

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