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Yoğunluklu Öklidyen 3-Uzayında Helis Yüzeyleri

Yıl 2021, Sayı: 21, 424 - 427, 31.01.2021
https://doi.org/10.31590/ejosat.798542

Öz

Farklı uzaylarda helis eğrilerinin ve helis hiperyüzeylerinin diferensiyel geometrisi, birçok bilim dalında önemli uygulama alanlarına sahiptir. Ayrıca, ağırlıklı manifold kavramı son yıllarda bilim insanları için çok popular bir konu olmaya başlamıştır. Bu bağlamda, yoğunluklu manifoldlar üzerinde n-boyutlu bir hiperyüzeyin ağırlıklı ortalama eğriliği (veya φ-ortalama eğriliği) ve ağırlıklı Gaussian eğriliği (veya φ-Gaussian eğriliği) kavramları tanımlandıktan sonra, farklı yoğunluğa sahip değişik uzaylarda diferensiyel geometriciler tarafından pek çok çalışma yapılmaktadır. Bu sebeple, biz de bu çalışmada, ilk olarak Öklidyen 3-uzayında bir helis yüzeyinin normal vektör alanını, ortalama eğriliğini ve Gaussian eğriliğini verdik ve ardından üç farklı yoğunluğa sahip üç boyutlu Öklidyen uzayında birim hızlı bir düzlemsel eğri tarafından oluşturulan bir helis yüzeyinin, parametrik denklemini ifade ederek, ağırlıklı ortalama eğriliğini ve ağırlıklı Gaussian eğriliğini elde ettik. Bununla birlikte, biliyoruz ki yoğunluklu Öklidyen 3-uzayında bir hiperyüzey ağırlıklı minimal ve ağırlıklı flattır, eğer sırasıyla ağırlıklı ortalama eğriliği ve ağırlıklı Gaussian eğriliği sıfır oluyorsa. Dolayısıyla, bu tanımları kullanarak, Öklidyen 3-uzayında bu farklı yoğunluklar için ağırlıklı minimal helis yüzeyleri elde ettik ve bu yüzeylerin ağırlıklı flatlığı için bazı sonuçlar verdik. Umuyoruz ki, bu çalışma sabit açılı yüzeyler ile ilgilenen diferensiyel geometricilere yeni bir bakış açısı kazandıracak ve yakın gelecekte değişik yoğunluklara sahip farklı uzaylarda bu yüzeyler ele alınabilecektir.

Kaynakça

  • Altın, M., & Kazan, A., & Karadağ, HB. (2019). Rotational Surfaces Generated by Planar Curves in E^3 with Density. Int J Anal Appl., (17(3)), 311-328.
  • Altın, M., & Kazan, A., & Karadağ, HB. (2019). Ruled Surfaces in E^3 with Density. Honam Mathematical Journal, (41(4)), 683-695.
  • Altın, M., & Kazan, A., & Karadağ, HB. (2020). Non-Null Curves with Constant Weighted Curvature in Lorentz-Minkowski Plane with Density. Turk J Math., (44), 588 – 610.
  • Altın, M., & Kazan, A., & Karadağ, HB. (2020). Ruled Surfaces Constructed by Planar Curves in Euclidean 3-Space with Density. Celal Bayar Univ Journal of Science, (16(1)), 81-88.
  • Cadena, CB., & Di Scala, AJ., & Ruiz-Hernandez, G. (2015). Helix surfaces in Euclidean spaces. Beitr Algebra Geom., (56), 551–573.
  • Corwin, I., & Hoffman, N., & Hurder, S., & Sesum, V., & Xu, Y. (2006). Differential geometry of manifolds with density. Rose-Hulman Und Math J., (7(1)), 1-15.
  • Di Scala, AJ., & Ruiz-Hernandez, G. (2009). Helix submanifolds of euclidean spaces. Monatsh Math., (157), 205–215.
  • Di Scala, AJ., & Ruiz-Hernandez, G. (2016). Minimal helix submanifolds and minimal Riemannian foliations. Bol Soc Mat Mex., (22), 229–250.
  • Di Scala, AJ. (2009). Weak helix submanifolds of euclidean spaces. Abh Math Semin Univ Hambg., (79), 37–46.
  • Fetcu, D. (2015). A classification result for helix surfaces with parallel mean curvature in product spaces. Ark Mat., (53), 249–258.
  • Gromov, M. (2003). Isoperimetry of waists and concentration of maps. Geom Func Anal., (13), 178-215.
  • Hieu, DT., & Hoang, NM. (2009). Ruled minimal surfaces in R^3 with density e^z. Pacific J. Math., (243(2)), 277-285.
  • Hieu, DT., & Nam, TL. (2013). The classification of constant weighted curvature curves in the plane with a log-linear density. Commun Pure Appl Anal., (13), 1641-1652.
  • Kazan, A., & Karadag, HB. (2018). Weighted Minimal and Weighted Flat Surfaces of Revolution in Galilean 3-Space with Density. Int J Anal Appl., (16(3)), 414-426.
  • Morgan, F. (2005). Manifolds with Density. Not Amer Math Soc., (52(8)), 853-858.
  • Nam, TL. (2017). Some results on curves in the plane with log-linear density. Asian-European J of Math., (10(2)), 1-8.
  • Ruiz-Hernandez, G. (2011). Minimal helix surfaces in N^n×R. Abh Math Semin Univ Hambg., (81), 55–67.
  • Yoon, DW., & Yüzbaşı, ZK. (2018). Weighted Minimal Affine Translation Surfaces in Euclidean Space with Density. International Journal of Geometric Methods in Modern Physics, (15(11)).
  • Zıplar, E., & Şenol, A., & Yaylı, Y. (2012). Helix Hypersurfaces and Special Curves. Int J Contemp Math Sciences, (7(25)), 1233-1245.
  • Zıplar, E. (2013). On the r-Helix Submanifolds of Euclidean Spaces. Ph.D.Thesis. Ankara University.

Helix Surfaces in Euclidean 3-Space with Density

Yıl 2021, Sayı: 21, 424 - 427, 31.01.2021
https://doi.org/10.31590/ejosat.798542

Öz

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 φ-mean curvature) and weighted Gaussian curvature (or φ-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.

Kaynakça

  • Altın, M., & Kazan, A., & Karadağ, HB. (2019). Rotational Surfaces Generated by Planar Curves in E^3 with Density. Int J Anal Appl., (17(3)), 311-328.
  • Altın, M., & Kazan, A., & Karadağ, HB. (2019). Ruled Surfaces in E^3 with Density. Honam Mathematical Journal, (41(4)), 683-695.
  • Altın, M., & Kazan, A., & Karadağ, HB. (2020). Non-Null Curves with Constant Weighted Curvature in Lorentz-Minkowski Plane with Density. Turk J Math., (44), 588 – 610.
  • Altın, M., & Kazan, A., & Karadağ, HB. (2020). Ruled Surfaces Constructed by Planar Curves in Euclidean 3-Space with Density. Celal Bayar Univ Journal of Science, (16(1)), 81-88.
  • Cadena, CB., & Di Scala, AJ., & Ruiz-Hernandez, G. (2015). Helix surfaces in Euclidean spaces. Beitr Algebra Geom., (56), 551–573.
  • Corwin, I., & Hoffman, N., & Hurder, S., & Sesum, V., & Xu, Y. (2006). Differential geometry of manifolds with density. Rose-Hulman Und Math J., (7(1)), 1-15.
  • Di Scala, AJ., & Ruiz-Hernandez, G. (2009). Helix submanifolds of euclidean spaces. Monatsh Math., (157), 205–215.
  • Di Scala, AJ., & Ruiz-Hernandez, G. (2016). Minimal helix submanifolds and minimal Riemannian foliations. Bol Soc Mat Mex., (22), 229–250.
  • Di Scala, AJ. (2009). Weak helix submanifolds of euclidean spaces. Abh Math Semin Univ Hambg., (79), 37–46.
  • Fetcu, D. (2015). A classification result for helix surfaces with parallel mean curvature in product spaces. Ark Mat., (53), 249–258.
  • Gromov, M. (2003). Isoperimetry of waists and concentration of maps. Geom Func Anal., (13), 178-215.
  • Hieu, DT., & Hoang, NM. (2009). Ruled minimal surfaces in R^3 with density e^z. Pacific J. Math., (243(2)), 277-285.
  • Hieu, DT., & Nam, TL. (2013). The classification of constant weighted curvature curves in the plane with a log-linear density. Commun Pure Appl Anal., (13), 1641-1652.
  • Kazan, A., & Karadag, HB. (2018). Weighted Minimal and Weighted Flat Surfaces of Revolution in Galilean 3-Space with Density. Int J Anal Appl., (16(3)), 414-426.
  • Morgan, F. (2005). Manifolds with Density. Not Amer Math Soc., (52(8)), 853-858.
  • Nam, TL. (2017). Some results on curves in the plane with log-linear density. Asian-European J of Math., (10(2)), 1-8.
  • Ruiz-Hernandez, G. (2011). Minimal helix surfaces in N^n×R. Abh Math Semin Univ Hambg., (81), 55–67.
  • Yoon, DW., & Yüzbaşı, ZK. (2018). Weighted Minimal Affine Translation Surfaces in Euclidean Space with Density. International Journal of Geometric Methods in Modern Physics, (15(11)).
  • Zıplar, E., & Şenol, A., & Yaylı, Y. (2012). Helix Hypersurfaces and Special Curves. Int J Contemp Math Sciences, (7(25)), 1233-1245.
  • Zıplar, E. (2013). On the r-Helix Submanifolds of Euclidean Spaces. Ph.D.Thesis. Ankara University.
Toplam 20 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Mühendislik
Bölüm Makaleler
Yazarlar

Ahmet Kazan 0000-0002-1959-6102

Sema Kazan 0000-0002-8771-9506

Yayımlanma Tarihi 31 Ocak 2021
Yayımlandığı Sayı Yıl 2021 Sayı: 21

Kaynak Göster

APA Kazan, A., & Kazan, S. (2021). Helix Surfaces in Euclidean 3-Space with Density. Avrupa Bilim Ve Teknoloji Dergisi(21), 424-427. https://doi.org/10.31590/ejosat.798542