The Gauss Map and the Third Laplace-Beltrami Operator of the Rotational Hypersurface in 4-Space
Abstract
:1. Introduction
2. Curvatures in
3. Rotational Hypersurface in
4. Gauss Map
5. The Third Laplace–Beltrami Operator
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
- Arslan, K.; Deszcz, R.; Yaprak, Ş. On Weyl pseudosymmetric hypersurfaces. Colloq. Math. 1997, 72, 353–361. [Google Scholar] [CrossRef]
- Arslan, K.; Kılıç Bayram, B.; Bulca, B.; Öztürk, G. Generalized Rotation Surfaces in . Results Math. 2012, 61, 315–327. [Google Scholar] [CrossRef]
- Arvanitoyeorgos, A.; Kaimakamis, G.; Magid, M. Lorentz hypersurfaces in satisfying ΔH = αH. Ill. J. Math. 2009, 53, 581–590. [Google Scholar]
- Chen, B.Y. Total Mean Curvature and Submanifolds of Finite Type; World Scientific: Singapore, 1984. [Google Scholar]
- Chen, B.Y.; Choi, M.; Kim, Y.H. Surfaces of revolution with pointwise 1-type Gauss map. Korean Math. Soc. 2005, 42, 447–455. [Google Scholar] [CrossRef]
- Dursun, U.; Turgay, N.C. Minimal and pseudo-umbilical rotational surfaces in Euclidean space . Mediterr. J. Math. 2013, 10, 497–506. [Google Scholar] [CrossRef]
- Kim, Y.H.; Turgay, N.C. Surfaces in with L1-pointwise 1-type Gauss map. Bull. Korean Math. Soc. 2013, 50, 935–949. [Google Scholar] [CrossRef]
- Takahashi, T. Minimal immersions of Riemannian manifolds. J. Math. Soc. Jpn. 1966, 18, 380–385. [Google Scholar] [CrossRef]
- Magid, M.; Scharlach, C.; Vrancken, L. Affine umbilical surfaces in . Manuscr. Math. 1995, 88, 275–289. [Google Scholar] [CrossRef]
- Vlachos, T. Hypersurfaces in with harmonic mean curvature vector field. Math. Nachr. 1995, 172, 145–169. [Google Scholar]
- Scharlach, C. Affine Geometry of Surfaces and Hypersurfaces in . In Symposium on the Differential Geometry of Submanifolds; Dillen, F., Simon, U., Vrancken, L., Eds.; Un. Valenciennes: Valenciennes, France, 2007; Volume 124, pp. 251–256. [Google Scholar]
- Cheng, Q.M.; Wan, Q.R. Complete hypersurfaces of with constant mean curvature. Monatsh. Math. 1994, 118, 171–204. [Google Scholar] [CrossRef]
- Moore, C. Surfaces of rotation in a space of four dimensions. Ann. Math. 1919, 21, 81–93. [Google Scholar] [CrossRef]
- Moore, C. Rotation surfaces of constant curvature in space of four dimensions. Bull. Am. Math. Soc. 1920, 26, 454–460. [Google Scholar] [CrossRef]
- Ganchev, G.; Milousheva, V. General rotational surfaces in the 4-dimensional Minkowski space. Turk. J. Math. 2014, 38, 883–895. [Google Scholar] [CrossRef] [Green Version]
- Dillen, F.; Fastenakels, J.; Van der Veken, J. Rotation hypersurfaces of and . Note Mat. 2009, 29-1, 41–54. [Google Scholar]
- Senoussi, B.; Bekkar, M. Helicoidal surfaces with ΔJr = Ar in 3-dimensional Euclidean space. Stud. Univ. Babeş-Bolyai Math. 2015, 60, 437–448. [Google Scholar]
- Bour, E. Théorie de la déformation des surfaces. J. Êcole Imp. Polytech. 1862, 22, 1–148. [Google Scholar]
- Do Carmo, M.; Dajczer, M. Helicoidal surfaces with constant mean curvature. Tohoku Math. J. 1982, 34, 351–367. [Google Scholar] [CrossRef]
- Güler, E. Bour’s theorem and light-like profile curve. Yokohama Math. J. 2007, 54, 155–157. [Google Scholar]
- Hieu, D.T.; Thang, N.N. Bour’s theorem in 4-dimensional Euclidean space. Bull. Korean Math. Soc. 2017, 54, 2081–2089. [Google Scholar]
- Choi, M.; Kim, Y.H.; Liu, H.; Yoon, D.W. Helicoidal surfaces and their Gauss map in Minkowski 3-space. Bull. Korean Math. Soc. 2010, 47, 859–881. [Google Scholar] [CrossRef]
- Choi, M.; Kim, Y.H. Characterization of the helicoid as ruled surfaces with pointwise 1-type Gauss map. Bull. Korean Math. Soc. 2001, 38, 753–761. [Google Scholar]
- Choi, M.; Yoon, D.W. Helicoidal surfaces of the third fundamental form in Minkowski 3-space. Bull. Korean Math. Soc. 2015, 52, 1569–1578. [Google Scholar] [CrossRef]
- Ferrandez, A.; Garay, O.J.; Lucas, P. On a Certain Class of Conformally at Euclidean Hypersurfaces. In Global Analysis and Global Differential Geometry; Springer: Berlin, Germany, 1990; pp. 48–54. [Google Scholar]
- Verstraelen, L.; Valrave, J.; Yaprak, Ş. The minimal translation surfaces in Euclidean space. Soochow J. Math. 1994, 20, 77–82. [Google Scholar]
- Güler, E.; Magid, M.; Yaylı, Y. Laplace Beltrami operator of a helicoidal hypersurface in four space. J. Geom. Symmetry Phys. 2016, 41, 77–95. [Google Scholar]
- Lawson, H.B. Lectures on Minimal Submanifolds, 2nd ed.; Mathematics Lecture Series 9; Publish or Perish, Inc.: Wilmington, Delaware, 1980. [Google Scholar]
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Güler, E.; Hacısalihoğlu, H.H.; Kim, Y.H. The Gauss Map and the Third Laplace-Beltrami Operator of the Rotational Hypersurface in 4-Space. Symmetry 2018, 10, 398. https://doi.org/10.3390/sym10090398
Güler E, Hacısalihoğlu HH, Kim YH. The Gauss Map and the Third Laplace-Beltrami Operator of the Rotational Hypersurface in 4-Space. Symmetry. 2018; 10(9):398. https://doi.org/10.3390/sym10090398
Chicago/Turabian StyleGüler, Erhan, Hasan Hilmi Hacısalihoğlu, and Young Ho Kim. 2018. "The Gauss Map and the Third Laplace-Beltrami Operator of the Rotational Hypersurface in 4-Space" Symmetry 10, no. 9: 398. https://doi.org/10.3390/sym10090398
APA StyleGüler, E., Hacısalihoğlu, H. H., & Kim, Y. H. (2018). The Gauss Map and the Third Laplace-Beltrami Operator of the Rotational Hypersurface in 4-Space. Symmetry, 10(9), 398. https://doi.org/10.3390/sym10090398