DINI-TYPE HELICOIDAL HYPERSURFACE IN 4-SPACE

Yıl 2019, Cilt: 1 Sayı: 1, 26 - 34, 18.01.2019

Erhan Güler Ayçın Gümüşok Karaalp

Öz

We define Dini-type helicoidal hypersurface in the four dimensional Euclidean space E4. We calculate the Gauss map, Gaussian curvature and the
mean curvature of the helicoidal hypersurface. Additionally, we find some special relations and symmetries for the curvatures.

Anahtar Kelimeler

Four dimensional Euclidean space, Gauss map, Dini-type helicoidal hypersurface

Kaynakça

• Aksoyak F., Yaylı Y. (2014) Boost invariant surfaces with pointwise 1-type Gauss map in Minkowski 4-Space E_1^4. Bull. Korean Math. Soc. 51: 1863-1874.
• Aksoyak F., Yaylı Y. (2015) General rotational surfaces with pointwise 1-type Gauss map in pseudo-Euclidean space E_2^4. Indian J. Pure Appl. Math. 46: 107-118.
• Arslan K., Bulca B., Milousheva V. (2014) Meridian surfaces in E^4 with pointwise 1-type Gauss map. Bull. Korean Math. Soc. 51: 911-922.
• Arslan K., Deszcz R., Yaprak Ş. (1997) On Weyl pseudosymmetric hypersurfaces. Colloq. Math. 72(2): 353-361.
• Arvanitoyeorgos A., Kaimakamis G., Magid M. (2009) Lorentz hypersurfaces in E_1^4 satisfying ΔH=αH. Illinois J. Math. 53(2): 581-590.
• Chen B.Y. (2014) Total Mean Curvature and Submanifolds of Finite Type. 2nd Edn, World Scientific, Hackensack.
• Chen B.Y., Choi M., Kim Y.H. (2005) Surfaces of revolution with pointwise 1-type Gauss map. Korean Math. Soc. 42: 447-455.
• Chen B.Y., Ishikawa S. (1993) On classification of some surfaces of revolution of finite type. Tsukuba J. Math.17(1): 287-298.
• Chen B.Y., Piccinni P. (1987) Submanifolds with finite type Gauss map. Bull. Aust. Math. Soc. 35: 161-186.
• Cheng Q.M., Wan Q.R. (1994) Complete hypersurfaces of R^4 with constant mean curvature. Monatsh. Math. 118: 171-204.
• Choi M., Kim Y.H. (2001) Characterization of the helicoid as ruled surfaces with pointwise 1-type Gauss map. Bull. Korean Math. Soc. 38: 753-761.
• Dillen F., Pas J., Verstraelen L. (1990) On surfaces of finite type in Euclidean 3-space. Kodai Math. J. 13: 10-21.
• Dini U. (1871) Sopra le funzioni di una variabile complessa, Annali di matematica pura ed applicata, 4(2): 159-174.
• Do Carmo M., Dajczer M. (1982) Helicoidal surfaces with constant mean curvature. Tohoku Math. J. 34: 351-367.
• Dursun U. (2009) Hypersurfaces with pointwise 1-type Gauss map in Lorentz-Minkowski space. Proc. Est. Acad. Sci. 58: 146-161.
• Dursun U., Turgay N.C. (2013) Minimal and pseudo-umbilical rotational surfaces in Euclidean space E^4. Mediterr. J. Math. 10: 497-506.
• Ferrandez A., Garay O.J., Lucas P. (1990) On a certain class of conformally at Euclidean hypersurfaces. In Global Analysis and Global Differential Geometry; Springer: 48-54, Berlin, Germany.
• Ganchev G., Milousheva V. (2014) General rotational surfaces in the 4-dimensional Minkowski space. Turkish J. Math. 38: 883-895.
• Güler 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): 1-11.
• Güler E., Kişi Ö. (2019) Dini-type helicoidal hypersurfaces with timelike axis in Minkowski 4-space E_1^4. Mathematics: Comp. Alg. Sci. Comp. 7(2): 1-8.
• Güler E., Magid M., Yaylı Y. (2016) Laplace Beltrami operator of a helicoidal hypersurface in four space. J. Geom. Sym. Phys. 41: 77-95.
• Güler E., Yaylı Y., Hacısalihoğlu H.H. (2010) Bour's theorem on the Gauss map in 3-Euclidean space. Hacettepe J. Math. Stat. 39: 515-525.
• Kim D.S., Kim J.R., Kim Y.H. (2016) Cheng-Yau operator and Gauss map of surfaces of revolution. Bull. Malays. Math. Sci. Soc. 39: 1319-1327.
• Kim Y.H., Turgay N.C. (2013) Surfaces in E^4 with L_1-pointwise 1-type Gauss map. Bull. Korean Math. Soc. 50(3): 935-949.
• Lawson H.B. (1980) Lectures on Minimal Submanifolds, 2nd ed.; Mathematics Lecture Series 9; Publish or Perish, Inc.: Wilmington, Delaware.
• Magid M., Scharlach C., Vrancken L. (1995) Affine umbilical surfaces in R^4. Manuscripta Math. 88: 275-289.
• Moore C. (1919) Surfaces of rotation in a space of four dimensions. Ann. Math. 21: 81-93.
• Moore C. (1920) Rotation surfaces of constant curvature in space of four dimensions. Bull. Amer. Math. Soc. 26: 454-460.
• Moruz M., Munteanu M.I. (2016) Minimal translation hypersurfaces in E^4. J. Math. Anal. Appl. 439: 798-812.
• Scharlach, C. (2007) Affine geometry of surfaces and hypersurfaces in R^4. In Symposium on the Differential Geometry of Submanifolds; Dillen F., Simon U., Vrancken L., Eds.; Un. Valenciennes: Valenciennes, France, 124: 251-256.
• Senoussi B., Bekkar M. (2015) Helicoidal surfaces with Δ^J r=Ar in 3-dimensional Euclidean space. Stud. Univ. Babeş-Bolyai Math. 60(3): 437-448.
• Takahashi T. (1966) Minimal immersions of Riemannian manifolds. J. Math. Soc. Japan 18: 380-385.
• Verstraelen L., Walrave J., Yaprak S. (1994) The minimal translation surfaces in Euclidean space. Soochow J. Math. 20(1): 77-82.
• Vlachos Th. (1995) Hypersurfaces in E^4 with harmonic mean curvature vector field. Math. Nachr. 172: 145-169.
• Yoon D.W. (2001) Rotation Surfaces with finite type Gauss map in E^4. Indian J. Pure Appl. Math. 32: 1803-1808.