Rotational Hypersurfaces Satisfying ∆^I R=AR in the Four-Dimensional Euclidean Space

Erhan Güler

doi:10.2339/politeknik.670333

Araştırma Makalesi

Rotational Hypersurfaces Satisfying ∆^I R=AR in the Four-Dimensional Euclidean Space

Yıl 2021, Cilt: 24 Sayı: 2, 517 - 520, 01.06.2021

Erhan Güler

https://doi.org/10.2339/politeknik.670333

Cited By: 5

Öz

In this study, rotational hypersurfaces in the 4-dimensional Euclidean space are discussed. Some relations of curvatures of hypersurfaces are given, such as the mean, Gaussian, and their minimality and flatness. In addition, Laplace-Beltrami operator has been defined for 4-dimensional hypersurfaces depending on the first fundamental form. Moreover, it is shown that each element of the 4×4 order matrix A, which satisfies the condition ∆^I R=AR, is zero, that is, the rotational hypersurface R is minimal.

Anahtar Kelimeler

4-dimensional Euclidean space, Laplace-Beltrami operator, rotational hypersurface, curvature

Kaynakça

[1] K. Arslan, B. Kılıç Bayram, B. Bulca, G. Öztürk, “Generalized Rotation Surfaces in E4,” Result Math. vol. 61, pp. 315–327, 2012.
[2] Arvanitoyeorgos, G. Kaimakamis, M. Magid “Lorentz hypersurfaces in E4,1 satisfying ,” Illinois J. Math. vol. 53, no. 2, pp. 581–590, 2009.
[3] E. Bour, “Théorie de la déformation des surfaces,” J. Êcole Imperiale Polytech., vol. 22, no. 39, pp. 1–148, 1862.
[4] B.Y. Chen, “Total mean curvature and submanifolds of finite type,” World Scientific, Singapore, 1984.
[5] B.Y. Chen, M. Choi, Y.H. Kim, “Surfaces of revolution with pointwise 1-type Gauss map,” Korean Math. Soc., vol. 42, pp. 447–455, 2005.
[6] Q.M. Cheng, Q.R. Wan, “Complete hypersurfaces of R4 with constant mean curvature,” Monatsh. Math. vol. 118, no. 3, pp. 171–204, 1994.
[7] M. Choi, Y.H. Kim, “Characterization of the helicoid as ruled surfaces with pointwise 1-type Gauss map,” Bull. Korean Math. Soc. vol. 38, pp. 753–761, 2001.
[8] F. Dillen, J. Pas, L. Verstraelen, “On surfaces of finite type in Euclidean 3-space,” Kodai Math. J. vol. 13, pp. 10–21, 1990.
[9] M. Do Carmo, M. Dajczer, “Helicoidal surfaces with constant mean curvature,” Tohoku Math. J. vol. 34, pp. 351–367, 1982.
[10] U. Dursun, N.C. Turgay, “Minimal and Pseudo-Umbilical Rotational Surfaces in Euclidean Space E4,” Mediter. J. Math. vol. 10, pp. 497–506, 2013.
[11] A. Ferrandez, O.J. Garay, P. Lucas, “On a certain class of conformally at Euclidean hypersurfaces,” In Global Analysis and Global Differential Geometry; Springer: Berlin, Germany, pp. 48–54. 1990.
[12] G. Ganchev, V. Milousheva, “General rotational surfaces in the 4-dimensional Minkowski space,” Turkish J. Math. vol. 38 pp. 883–895, 2014.
[13] E. Güler, “Bour's theorem and lightlike profile curve,” Yokohama Math. J., vol. 54, no. 1, pp. 55–77, 2007.
[14] E. Güler, M. Magid, Y. Yaylı, “Laplace Beltrami operator of a helicoidal hypersurface in four space,” J. Geom. Symmetry Phys. vol. 41, pp. 77–95, 2016.
[15] E. Güler, H.H. Hacısalihoğlu, Y.H. Kim The Gauss map and the third Laplace-Beltrami operator of the rotational hypersurface in 4-Space. Symmetry, 10(9), 398, 2018.
[16] H.B. Lawson, “Lectures on Minimal Submanifolds,” 2nd ed.; Mathematics Lecture Series 9; Publish or Perish, Inc.: Wilmington, Delaware, 1980.
[17] M. Magid, C. Scharlach, L. Vrancken, “Affine umbilical surfaces in R4,” Manuscripta Math. vol. 88, pp. 275–289, 1995.
[18] C. Moore, “Surfaces of rotation in a space of four dimensions,” Ann. Math., vol. 21, pp. 81–93, 1919.
[19] C. Moore, “Rotation surfaces of constant curvature in space of four dimensions,” Bull. Amer. Math. Soc., vol. 26, pp. 454–460, 1920.
[20]M. Moruz, M.I. Munteanu, “Minimal translation hypersurfaces in E4,” J. Math. Anal. Appl., vol. 439, pp. 798–812, 2016.
[21] C. Scharlach, “Affine geometry of surfaces and hypersurfaces in R4,” In Symposium on the Differential Geometry of Submanifolds; Dillen, F., Simon, U., Vrancken, L., Eds.; Un. Valenciennes: Valenciennes, France, 2007; V. 124, pp. 251–256.
[22] B. Senoussi, M. Bekkar, “Helicoidal surfaces with in 3-dimensional Euclidean space,” Stud. Univ. Babeş-Bolyai Math., vol. 60, no. 3, pp. 437–448, 2015.
[23] T. Takahashi, “Minimal immersions of Riemannian manifolds,” J. Math. Soc. Japan, vol. 18, pp. 380–385, 1966.
[24] L. Verstraelen, J. Walrave, Ş. Yaprak, “The minimal translation surfaces in Euclidean space,“ Soochow J. Math., vol. 20, no. 1, pp. 77–82, 1994.
[25] TH. Vlachos, “Hypersurfaces in E4 with harmonic mean curvature vector field,” Math. Nachr., vol. 172, pp. 145–169, 1995.

Rotational Hypersurfaces Satisfying ∆^I R=AR in the Four-Dimensional Euclidean Space

Yıl 2021, Cilt: 24 Sayı: 2, 517 - 520, 01.06.2021

Erhan Güler

https://doi.org/10.2339/politeknik.670333

Cited By: 5

Öz

Anahtar Kelimeler

Laplace-Beltrami operator, rotational hypersurface, 4-dimensional Euclidean space, curvature

Kaynakça

[1] K. Arslan, B. Kılıç Bayram, B. Bulca, G. Öztürk, “Generalized Rotation Surfaces in E4,” Result Math. vol. 61, pp. 315–327, 2012.
[2] Arvanitoyeorgos, G. Kaimakamis, M. Magid “Lorentz hypersurfaces in E4,1 satisfying ,” Illinois J. Math. vol. 53, no. 2, pp. 581–590, 2009.
[3] E. Bour, “Théorie de la déformation des surfaces,” J. Êcole Imperiale Polytech., vol. 22, no. 39, pp. 1–148, 1862.
[4] B.Y. Chen, “Total mean curvature and submanifolds of finite type,” World Scientific, Singapore, 1984.
[5] B.Y. Chen, M. Choi, Y.H. Kim, “Surfaces of revolution with pointwise 1-type Gauss map,” Korean Math. Soc., vol. 42, pp. 447–455, 2005.
[6] Q.M. Cheng, Q.R. Wan, “Complete hypersurfaces of R4 with constant mean curvature,” Monatsh. Math. vol. 118, no. 3, pp. 171–204, 1994.
[7] M. Choi, Y.H. Kim, “Characterization of the helicoid as ruled surfaces with pointwise 1-type Gauss map,” Bull. Korean Math. Soc. vol. 38, pp. 753–761, 2001.
[8] F. Dillen, J. Pas, L. Verstraelen, “On surfaces of finite type in Euclidean 3-space,” Kodai Math. J. vol. 13, pp. 10–21, 1990.
[9] M. Do Carmo, M. Dajczer, “Helicoidal surfaces with constant mean curvature,” Tohoku Math. J. vol. 34, pp. 351–367, 1982.
[10] U. Dursun, N.C. Turgay, “Minimal and Pseudo-Umbilical Rotational Surfaces in Euclidean Space E4,” Mediter. J. Math. vol. 10, pp. 497–506, 2013.
[11] A. Ferrandez, O.J. Garay, P. Lucas, “On a certain class of conformally at Euclidean hypersurfaces,” In Global Analysis and Global Differential Geometry; Springer: Berlin, Germany, pp. 48–54. 1990.
[12] G. Ganchev, V. Milousheva, “General rotational surfaces in the 4-dimensional Minkowski space,” Turkish J. Math. vol. 38 pp. 883–895, 2014.
[13] E. Güler, “Bour's theorem and lightlike profile curve,” Yokohama Math. J., vol. 54, no. 1, pp. 55–77, 2007.
[14] E. Güler, M. Magid, Y. Yaylı, “Laplace Beltrami operator of a helicoidal hypersurface in four space,” J. Geom. Symmetry Phys. vol. 41, pp. 77–95, 2016.
[15] E. Güler, H.H. Hacısalihoğlu, Y.H. Kim The Gauss map and the third Laplace-Beltrami operator of the rotational hypersurface in 4-Space. Symmetry, 10(9), 398, 2018.
[16] H.B. Lawson, “Lectures on Minimal Submanifolds,” 2nd ed.; Mathematics Lecture Series 9; Publish or Perish, Inc.: Wilmington, Delaware, 1980.
[17] M. Magid, C. Scharlach, L. Vrancken, “Affine umbilical surfaces in R4,” Manuscripta Math. vol. 88, pp. 275–289, 1995.
[18] C. Moore, “Surfaces of rotation in a space of four dimensions,” Ann. Math., vol. 21, pp. 81–93, 1919.
[19] C. Moore, “Rotation surfaces of constant curvature in space of four dimensions,” Bull. Amer. Math. Soc., vol. 26, pp. 454–460, 1920.
[20]M. Moruz, M.I. Munteanu, “Minimal translation hypersurfaces in E4,” J. Math. Anal. Appl., vol. 439, pp. 798–812, 2016.
[21] C. Scharlach, “Affine geometry of surfaces and hypersurfaces in R4,” In Symposium on the Differential Geometry of Submanifolds; Dillen, F., Simon, U., Vrancken, L., Eds.; Un. Valenciennes: Valenciennes, France, 2007; V. 124, pp. 251–256.
[22] B. Senoussi, M. Bekkar, “Helicoidal surfaces with in 3-dimensional Euclidean space,” Stud. Univ. Babeş-Bolyai Math., vol. 60, no. 3, pp. 437–448, 2015.
[23] T. Takahashi, “Minimal immersions of Riemannian manifolds,” J. Math. Soc. Japan, vol. 18, pp. 380–385, 1966.
[24] L. Verstraelen, J. Walrave, Ş. Yaprak, “The minimal translation surfaces in Euclidean space,“ Soochow J. Math., vol. 20, no. 1, pp. 77–82, 1994.
[25] TH. Vlachos, “Hypersurfaces in E4 with harmonic mean curvature vector field,” Math. Nachr., vol. 172, pp. 145–169, 1995.

Toplam 25 adet kaynakça vardır.

Ayrıntılar

Birincil Dil	İngilizce
Konular	Mühendislik
Bölüm	Araştırma Makalesi
Yazarlar	Erhan Güler 0000-0003-3264-6239
Yayımlanma Tarihi	1 Haziran 2021
Gönderilme Tarihi	4 Ocak 2020
Yayımlandığı Sayı	Yıl 2021 Cilt: 24 Sayı: 2

Kaynak Göster

APA	Güler, E. (2021). Rotational Hypersurfaces Satisfying ∆^I R=AR in the Four-Dimensional Euclidean Space. Politeknik Dergisi, 24(2), 517-520. https://doi.org/10.2339/politeknik.670333
AMA	Güler E. Rotational Hypersurfaces Satisfying ∆^I R=AR in the Four-Dimensional Euclidean Space. Politeknik Dergisi. Haziran 2021;24(2):517-520. doi:10.2339/politeknik.670333
Chicago	Güler, Erhan. “Rotational Hypersurfaces Satisfying ∆^I R=AR in the Four-Dimensional Euclidean Space”. Politeknik Dergisi 24, sy. 2 (Haziran 2021): 517-20. https://doi.org/10.2339/politeknik.670333.
EndNote	Güler E (01 Haziran 2021) Rotational Hypersurfaces Satisfying ∆^I R=AR in the Four-Dimensional Euclidean Space. Politeknik Dergisi 24 2 517–520.
IEEE	E. Güler, “Rotational Hypersurfaces Satisfying ∆^I R=AR in the Four-Dimensional Euclidean Space”, Politeknik Dergisi, c. 24, sy. 2, ss. 517–520, 2021, doi: 10.2339/politeknik.670333.
ISNAD	Güler, Erhan. “Rotational Hypersurfaces Satisfying ∆^I R=AR in the Four-Dimensional Euclidean Space”. Politeknik Dergisi 24/2 (Haziran 2021), 517-520. https://doi.org/10.2339/politeknik.670333.
JAMA	Güler E. Rotational Hypersurfaces Satisfying ∆^I R=AR in the Four-Dimensional Euclidean Space. Politeknik Dergisi. 2021;24:517–520.
MLA	Güler, Erhan. “Rotational Hypersurfaces Satisfying ∆^I R=AR in the Four-Dimensional Euclidean Space”. Politeknik Dergisi, c. 24, sy. 2, 2021, ss. 517-20, doi:10.2339/politeknik.670333.
Vancouver	Güler E. Rotational Hypersurfaces Satisfying ∆^I R=AR in the Four-Dimensional Euclidean Space. Politeknik Dergisi. 2021;24(2):517-20.