The Principal Curvatures and the Third Fundamental Form of Dini-Type Helicoidal Hypersurface in 4-Space
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Published
Dec 30, 2020
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62-68
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Erhan G¨uler
Bartın University, Faculty of Sciences, Department of Mathematics, 74100 Bartın, Turkey
Abstract
We consider the principal curvatures and the third fundamental form of Dini-type helicoidal hypersurface D(u, v, w) in the four dimensional Euclidean space E4. We find the Gauss map e of helicoidal hypersurface in E4. We obtain characteristic polynomial of shape operator matrix S. Then, we compute principal curvatures ki=1;2;3, and the third fundamental form matrix III of D.
Keywords:
Four dimensional, Dini-type helicoidal hypersurface, Gauss map, principal curvatures, the third fundamental form.
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Original Research Article
References
Arvanitoyeorgos A, Kaimakamis G, Magid M. Lorentz hypersurfaces in E4 1 satisfying ∆H = αH. Illinois J. Math. 2009;53(2):581-590.
Bour E. Theorie de la deformation des surfaces. J. de l. Ecole Imperiale Polytechnique. 1862;22(39):1-148.
Chen BY. Total mean curvature and submanifolds of finite type. World Scientific, Singapore; 1984.
Cheng QM, Wan QR. Complete hypersurfaces of R4 with constant mean curvature. Monatsh. Math. 1994;118(3-4):171-204.
Choi M, Kim YH. Characterization of the helicoid as ruled surfaces with pointwise 1-type Gauss map. Bull. Korean Math. Soc. 2001;38:753-761.
Dillen F, Pas J, Verstraelen L. On surfaces of finite type in Euclidean 3-space. Kodai Math. J. 1990;13:10-21.
Do Carmo M, Dajczer M. Helicoidal surfaces with constant mean curvature. Tohoku Math. J. 1982;34:351-367.
Ferrandez A, Garay OJ, Lucas P. On a certain class of conformally at Euclidean hypersurfaces. Proc. of the Conf. in Global Analysis and Global Differential Geometry, Berlin; 1990.
Ganchev G, Milousheva V. General rotational surfaces in the 4-dimensional Minkowski space. Turkish J. Math. 2014;38:883-895.
G¨uler E, Hacısaliho˘glu HH, Kim YH. The Gauss map and the third Laplace-Beltrami operator of the rotational hypersurface in 4-space. Symmetry. 2018;10(9):1-11.
G¨uler E, Ki¸si O. Dini-type helicoidal hypersurfaces with timelike axis in Minkowski 4-space ¨ E4 1. Mathematics. 2019;7(2)205:1-8.
G¨uler E, Magid M, Yaylı Y. Laplace Beltrami operator of a helicoidal hypersurface in four space. J. Geom. Sym. Phys. 2016;41:77-95.
Gler E, Turgay NC. Cheng-Yau operator and Gauss map of rotational hypersurfaces in 4-space. Mediterr. J. Math. 16(3) 66, (2019) 1–16.
G¨uler E, Yaylı Y, Hacısaliho˘glu HH. Bour’s theorem on the Gauss map in 3-Euclidean space. Hacettepe J. Math. 2010;39:515-525.
Hasanis Th, Vlachos Th. Hypersurfaces in E4 with harmonic mean curvature vector field. Math. Nachr. 1995;172:145-169.
Kim DS, Kim JR, Kim YH. Cheng-Yau operator and Gauss map of surfaces of revolution. Bull. Malays. Math. Sci. Soc. 2016;39:1319-1327.
Kim YH, Turgay NC. Surfaces in E4 with L1-pointwise 1-type Gauss map. Bull. Korean Math. Soc. 2013;50(3):935-949.
Lawson HB. Lectures on minimal submanifolds. Rio de Janeiro. 1973;1.
Magid M, Scharlach C, Vrancken L. Affine umbilical surfaces in R4. Manuscripta Math. 1995;88:275-289.
Moore C. Surfaces of rotation in a space of four dimensions. Ann. Math. 1919;21:81-93.
Moore C. Rotation surfaces of constant curvature in space of four dimensions. Bull. Amer. Math. Soc. 1920;26:454-460.
Moruz M, Munteanu MI. Minimal translation hypersurfaces in E4. J. Math. Anal. Appl. 2016;439:798-812.
Scharlach C. Affine geometry of surfaces and hypersurfaces in R4. Symposium on the Differential Geometry of Submanifolds, France. 2007;251-256.
Senoussi B, Bekkar M. Helicoidal surfaces with ∆Jr = Ar in 3-dimensional Euclidean space. Stud. Univ. Babe-Bolyai Math. 2015;60(3):437-448.
Takahashi T. Minimal immersions of Riemannian manifolds. J. Math. Soc. Japan. 1966;18:380- 385.
Verstraelen L, Walrave J, Yaprak S¸. The minimal translation surfaces in Euclidean space. Soochow J. Math. 1994;20(1):77-82
Bour E. Theorie de la deformation des surfaces. J. de l. Ecole Imperiale Polytechnique. 1862;22(39):1-148.
Chen BY. Total mean curvature and submanifolds of finite type. World Scientific, Singapore; 1984.
Cheng QM, Wan QR. Complete hypersurfaces of R4 with constant mean curvature. Monatsh. Math. 1994;118(3-4):171-204.
Choi M, Kim YH. Characterization of the helicoid as ruled surfaces with pointwise 1-type Gauss map. Bull. Korean Math. Soc. 2001;38:753-761.
Dillen F, Pas J, Verstraelen L. On surfaces of finite type in Euclidean 3-space. Kodai Math. J. 1990;13:10-21.
Do Carmo M, Dajczer M. Helicoidal surfaces with constant mean curvature. Tohoku Math. J. 1982;34:351-367.
Ferrandez A, Garay OJ, Lucas P. On a certain class of conformally at Euclidean hypersurfaces. Proc. of the Conf. in Global Analysis and Global Differential Geometry, Berlin; 1990.
Ganchev G, Milousheva V. General rotational surfaces in the 4-dimensional Minkowski space. Turkish J. Math. 2014;38:883-895.
G¨uler E, Hacısaliho˘glu HH, Kim YH. The Gauss map and the third Laplace-Beltrami operator of the rotational hypersurface in 4-space. Symmetry. 2018;10(9):1-11.
G¨uler E, Ki¸si O. Dini-type helicoidal hypersurfaces with timelike axis in Minkowski 4-space ¨ E4 1. Mathematics. 2019;7(2)205:1-8.
G¨uler E, Magid M, Yaylı Y. Laplace Beltrami operator of a helicoidal hypersurface in four space. J. Geom. Sym. Phys. 2016;41:77-95.
Gler E, Turgay NC. Cheng-Yau operator and Gauss map of rotational hypersurfaces in 4-space. Mediterr. J. Math. 16(3) 66, (2019) 1–16.
G¨uler E, Yaylı Y, Hacısaliho˘glu HH. Bour’s theorem on the Gauss map in 3-Euclidean space. Hacettepe J. Math. 2010;39:515-525.
Hasanis Th, Vlachos Th. Hypersurfaces in E4 with harmonic mean curvature vector field. Math. Nachr. 1995;172:145-169.
Kim DS, Kim JR, Kim YH. Cheng-Yau operator and Gauss map of surfaces of revolution. Bull. Malays. Math. Sci. Soc. 2016;39:1319-1327.
Kim YH, Turgay NC. Surfaces in E4 with L1-pointwise 1-type Gauss map. Bull. Korean Math. Soc. 2013;50(3):935-949.
Lawson HB. Lectures on minimal submanifolds. Rio de Janeiro. 1973;1.
Magid M, Scharlach C, Vrancken L. Affine umbilical surfaces in R4. Manuscripta Math. 1995;88:275-289.
Moore C. Surfaces of rotation in a space of four dimensions. Ann. Math. 1919;21:81-93.
Moore C. Rotation surfaces of constant curvature in space of four dimensions. Bull. Amer. Math. Soc. 1920;26:454-460.
Moruz M, Munteanu MI. Minimal translation hypersurfaces in E4. J. Math. Anal. Appl. 2016;439:798-812.
Scharlach C. Affine geometry of surfaces and hypersurfaces in R4. Symposium on the Differential Geometry of Submanifolds, France. 2007;251-256.
Senoussi B, Bekkar M. Helicoidal surfaces with ∆Jr = Ar in 3-dimensional Euclidean space. Stud. Univ. Babe-Bolyai Math. 2015;60(3):437-448.
Takahashi T. Minimal immersions of Riemannian manifolds. J. Math. Soc. Japan. 1966;18:380- 385.
Verstraelen L, Walrave J, Yaprak S¸. The minimal translation surfaces in Euclidean space. Soochow J. Math. 1994;20(1):77-82