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

Year 2021, , 517 - 520, 01.06.2021

Erhan Güler

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

Cited By: 6

Abstract

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.

Keywords

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

References

• [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.