Rotational Hypersurfaces in Euclidean 4-Space with Density

Year 2022, Volume: 25 Issue: 1, 107 - 114, 01.03.2022

Mustafa Altın

https://doi.org/10.2339/politeknik.740513

Cited By: 3

Abstract

In this paper, the Euclidean 4-space with a positive density function e^(x^2+y^2+z^2+t^2 ) is studied. Firstly, the weighted mean and weighted Gaussian curvature functions of a rotational hypersurface in 4-dimensional Euclidean space with density are obtained. The rotational hypersurfaces are constructed by solving these obtained functions which are second-order non-linear ordinary differential equations. Besides, the examples of rotational hypersurfaces are given with the aid of the weighted Gaussian and weighted mean curvatures in E^4 with density.

Keywords

Rotational hypersurfaces, Euclidean 4-space with density, weighted mean curvatures, weighted Gaussian curvatures

References

• [1] Moore, C., “Surfaces of rotation in a space of four dimensions”, Ann. Math., 21(2): 81-93, (1919).
• [2] Moore, C., “Rotation surfaces of constant curvature in space of four dimensions”, Bull. Amer. Math. Soc., 26(10): 454-460, (1920).
• [3] Cheng Q.M.and Wan, Q.R., “Complete hypersurfaces of R^4 with constant mean curvature”, Monatsh. Mth., 118(3-4): 171-204, (1994).
• [4] Yoon, D.W., “Rotation surfaces with finite type Gauss map in E^4”, Indian J. Pure Appl. Math., 32(12): 1803-1808, (2001).
• [5] Arslan, K., Kılıç, B, Bulca, B. and Öztürk, G., “Generalized Rotation Surfaces in E^4”, Results Math., 61(3-4): 315-327, (2012).
• [6] Arslan, K., Bayram, B., Bulca, B. and Öztürk, G., “On translation surfaces in 4-dimensional Euclidean space”, Acta et Com. Uni. Tar. De Math., 20(2):123-133, (2016).
• [7] Ganchev, G. and Milousheva, V., “General rotational surfaces in the 4-dimensional Minkowski space”, Turkish J. Math., 38: 883-895, (2014).
• [8] Moruz, M. and Monteanu, M.I., “Minimal translation hypersurfaces in E^4”, J. Math. Anal. Appl. 439(2): 798-812, (2016).
• [9] Dursun, U. and Turgay, N.C., “Minimal and pseudo-umbilical rotational surfaces in Euclidean space E^4”, Mediterr. J. Math., 10(1): 497-506, (2013).
• [10] Kahraman, F. and Yaylı, Y., “Boost invariant surface with pointwise 1-type Gauss map in Minkowski 4-space E_1^4”, Bull. Korean Math. Soc., 51: 1863-1874, (2014).
• [11] Kahraman, F. and Yaylı, Y., “General rotational surfaces with pointwise 1-type Gauss map in pseudo-Euclidean space E_2^4”, Indian J. Pure Appl. Math., 46: 107-118, (2014).
• [12] Güler, E., Magid, M. and Yaylı, Y., “Laplace-Beltrami operator of a helicoidal hypersurface in four space”, J. Goem. and Sym. Phys., 41: 77-95, (2016).
• [13] Güler, E., Hacısalihoµglu, H.H. and Kim, Y.H., “The Gauss map and the third Laplace-Beltrami operator of the rotational hypersurface in 4-space”, Symmetry, 10(398): 1-11, (2018).
• [14] M. Gromov, “Isoperimetry of waists and concentration of maps”, Geom. Func. Anal., 13: 178-215, (2003).
• [15] I. Corwin, N. Hoffman, S. Hurder, V. Sesum and Y. Xu, “Differential geometry of manifolds with density”, Rose-Hulman Und. Math. J., 7(1): 1-15, (2006).
• [16] L. Belarbi and M. Belkhelfa, “Surfaces in R^3 with Density”, i-manager.s Journal on Mathematics, 1(1): 34-48, (2012).
• [17] D.T. Hieu and T.L. Nam, “The classification of constant weighted curvature curves in the plane with a log-linear density”, Commun. Pure Appl. Anal., 13(4): 1641-1652, (2014).
• [18] M. Altın, A. Kazan and H.B. Karadağ, “Rotational surfaces Generated by Planar Curves in E^3 with Density”, International Journal of Analysis and Applications, 17(3): 311-328, (2019).
• [19] A. Kazan and H.B. Karadağ, “Weighted Minimal And Weighted Flat Surfaces of Revolution in Galilean 3-Space with Density”, Int. J. Anal. Appl., 16(3): 414-426, (2018). [20] F. Morgan, “Manifolds with Density”, Not. Amer. Math. Soc., 52(8): 853-858, (2005).
• [21] F. Morgan, “Myers’ Theorem With Density”, Kodai Math. J., 29: 455-461, (2006).
• [22] T.L. Nam, “Some results on curves in the plane with log-linear density”, Asian-European J. of Math., 10(2): 1-8, (2017).
• [23] D.W. Yoon, D-S. Kim, Y.H. Kim and J.W. Lee, “Constructions of Helicoidal Surfaces in Euclidean Space with Density”, Symmetry, 173: 1-9, (2017).
• [24] D.W. Yoon and Z.K. Yüzbaşı, “Weighted Minimal Affine Translation Surfaces in Euclidean Space with Density”, International Journal of Geometric Methods in Modern Physics, 15:11, (2018).
• [25] Ö.G. Yıldız, S. Hızal and M. Akyiğit, “Type I+ Helicoidal Surfaces with Prescribed Weighted Mean or Gaussian Curvature in Minkowski Space with Density”, An. S.t. Univ. Ovidius Constanta, 26(3): 99-108, (2018).
• [26] Belarbi L, Belkhelfa M. Some Results in Riemannian Manifolds with Density. Analele Universitatii din Oradea. Fascicola Matematica Tom XXII, 2: 81-86 (2015).
• [27] Hieu DT, Hoang NM. Ruled Minimal Surfaces in R^3 with Density e^z . Pacific Journal of Mathematics 243(2): 277-285, (2009).
• [28] López R. Minimal surfaces in Euclidean space with a log-linear density. arXiv:1410.2517v1 2014.
• [29] Morgan F. Manifolds with Density and Perelman’s Proof of the Poincare Conjecture. Mathematical Association of America, 116(2): 134-142, (2009).
• [30] Yoon DW. Weighted Minimal Translation Surfaces in Minkowski 3-space with Density. International Journal of Geometric Methods in Modern Physics, 14(12): 1-10, (2017)
• [31] Yoon DW. Weighted Minimal Translation Surfaces in the Galilean Space with Density. Open Mathematics; 15: 459-466, (2017).
• [32] Güler, E. Kişi Ö., “The Second Laplace-Beltrami operator on rotational hypersurface in the Euclidean 4-space”, Mathematica Aeterna, 8(1): 1-12, (2018).
• [33] Güler, E., Turgay N.C., “ Cheng-Yau operatör and Gauss map of the rotational hypersurface in 4-space”, Mediterranean Journal of Mathematics, 16(66): 1-16, (2019).
• [34] Yüce, S. “Weingarten Map of the Hypersurface in Euclidean 4-Space and its Applications” Hagia Sophia Journal of Geometry, 1:1, (2019).
• [35] Altin M., Kazan A. and Karadağ, H.B., “Non-Null Curves With Constant Weıghted Curvature In Lorentz-Minkowski Plane With Density”Turkish Journal of Mathematics”, 44(2), (2020).
• [36] Altin M., Kazan A. and Karadağ H.B. “Ruled Surfaces Constructed by Planar Curves in Euclidean 3-Space with Density”, “Celal Bayar University Journal of Science”, 16(2): 81-88, (2020)
• [37] Altin M., Kazan A., and Karadağ, H.B., “Monge Hypersurfaces in Euclidean 4 Space with Density” Journal of Polytechnic, 23: 207–214, (2020).
• [38] Altin M., Kazan A. and Karadağ, H.B., “Rotational Surfaces Generated by Planar Curves in E^3 with Density,”International Journal of Analysis and Applications”, 17(3): 311-328, (2019)