BI-ROTATIONAL HYPERSURFACE WITH Δx=Ax IN 4-SPACE

Erhan Güler, Yusuf Yaylı, Hasan Hilmi Hacısalihoğlu

DOI Number
https://doi.org/10.22190/FUMI211204063G
First page
917
Last page
928

Abstract


We introduce the bi-rotational hypersurface $\mathbf{x(}u,v,w\mathbf{)}$ in the four dimensional Euclidean geometry ${\mathbb{E}}^{4}.$ We obtain the $i$-th curvatures of the hypersurface. Moreover, we consider the Laplace--Beltrami operator of the bi-rotational hypersurface satisfying $ \Delta \mathbf{x=}\mathcal{A}\mathbf{x}$ for some $4\times 4$ matrix $ \mathcal{A}$.

Keywords

bi-rotational hypersurface, Eucledian geometry, curvative formulas

Full Text:

PDF

References


Alias L.J., Gürbüz N.: An extension of Takashi theorem for the linearized operators of the highest order mean curvatures, Geom. Dedicata 121, 113-127 (2006).

Aminov Y.: The geometry of submanifolds. Gordon and Breach Sci. Pub., Amsterdam, 2001.

Arslan K., Bayram B.K., Bulca B., Kim Y.H., Murathan C., Öztürk G.: Vranceanu surface in E4 with pointwise 1-type Gauss map. Indian J. Pure Appl. Math. 42(1), 41-51 (2011).

Arslan, K., Bayram B.K., Bulca B., Oztürk G.: Generalized rotation surfaces in E4. Results Math. 61(3), 315{327 (2012).

Arslan K., Bulca B., Kılıç B., Kim Y.H., Murathan C., Öztürk G.: Tensor product surfaces with pointwıse 1-type Gauss map. Bull. Korean Math. Soc. 48(3), 601-609 (2011).

Arslan K., Milousheva V.: Meridian surfaces of elliptic or hyperbolic type with pointwise 1-type Gauss map in Minkowski 4-space. Taiwanese J. Math. 20(2), 311{332 (2016).

Arslan K., Sütveren A., Bulca B.: Rotational λ-hypersurfaces in Euclidean spaces. Creat. Math. Inform. 30(1), 29{40 (2021).

Arvanitoyeorgos A., Kaimakamis G., Magid M.: Lorentz hypersurfaces in E4 1 satisfying ∆H = αH: Ill. J. Math. 53(2) 581-590 (2009).

Barros M., Chen B.Y.: Stationary 2-type surfaces in a hypersphere. J. Math. Soc. Jpn. 39(4), 627-648 (1987).

Barros M., Garay O.J.: 2-type surfaces in S3. Geom. Dedicata 24(3), 329-336 (1987).

Bektaş B., Canfes E.Ö., Dursun U.: Classification of surfaces in a pseudo-sphere with ¨

-type pseudo-spherical Gauss map. Math. Nachr. 290(16), 2512-2523 (2017).

Bektaş, B., Canfes E.O., Dursun U.: Pseudo-spherical submanifolds with 1-type pseudospherical Gauss map. Results Math. 71(3), 867-887 (2017).

Chen B.Y.: On submanifolds of finite type. Soochow J. Math. 9, 65-81 (1983).

Chen B.Y.: Total mean curvature and submanifolds of finite type. World Scientific, Singapore (1984).

Chen B.Y.: Finite type submanifolds and generalizations. University of Rome, 1985.

Chen B.Y.: Finite type submanifolds in pseudo-Euclidean spaces and applications. Kodai Math. J. 8(3), 358-374 (1985).

Chen B.Y., Piccinni, P.: Submanifolds with finite type Gauss map. Bull. Aust. Math. Soc. 35, 161-186 (1987).

Cheng Q.M., Wan Q.R.: Complete hypersurfaces of R4 with constant mean curvature. Monatsh. Math. 118, 171-204 (1994).

Cheng S.Y., Yau S.T.: Hypersurfaces with constant scalar curvature. Math. Ann. 225, 195-204 (1977).

Choi M., Kim Y.H.: Characterization of the helicoid as ruled surfaces with pointwise 1-type Gauss map. Bull. Korean Math. Soc. 38, 753-761 (2001).

Dillen F., Pas J., Verstraelen L.: On surfaces of finite type in Euclidean 3-space. Kodai Math. J. 13, 10-21 (1990).

Do Carmo M., Dajczer M.: Rotation hypersurfaces in spaces of constant curvature. Trans. Amer. Math. Soc. 277, 685-709 (1983).

Dursun U.: Hypersurfaces with pointwise 1-type Gauss map. Taiwanese J. Math. 11(5), 1407-1416 (2007).

Dursun U., Turgay N.C.: Space-like surfaces in Minkowski space E4 1 with pointwise 1-type Gauss map. Ukrainian Math. J. 71(1), 64-80 (2019).

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 48-54 (1990).

Ganchev G., Milousheva V.: General rotational surfaces in the 4-dimensional Minkowski space. Turk. J. Math. 38, 883-895 (2014).

Garay O.J.: On a certain class of finite type surfaces of revolution. Kodai Math. J. 11, 25-31 (1988).

Garay O.: An extension of Takahashi’s theorem. Geom. Dedicata 34, 105-112 (1990).

Güler E.: Fundamental form IV and curvature formulas of the hypersphere. Malaya J. Mat. 8(4), 2008-2011 (2020).

Güler E.: Rotational hypersurfaces satisfying ∆IR = AR in the four-dimensional Euclidean space. J. Polytech. 24(2), 517-520 (2021).

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 10(9), 1-12 (2018).

Güler E., Magid M., Yaylı Y.: Laplace{Beltrami operator of a helicoidal hypersurface in four-space. J. Geom. Symmetry Phys. 41, 77-95 (2016).

Güler E., Turgay N.C.: Cheng{Yau operator and Gauss map of rotational hypersurfaces in 4-space. Mediterr. J. Math. 16(3), 1-16 (2019).

Hasanis Th., Vlachos Th.: Hypersurfaces in E4 with harmonic mean curvature vector field. Math. Nachr. 172, 145-169 (1995).

Kahraman Aksoyak F., Yaylı Y.: Flat rotational surfaces with pointwise 1-type Gauss map in E4. Honam Math. J. 38(2), 305-316 (2016).

Kahraman Aksoyak F., Yaylı Y.: General rotational surfaces with pointwise 1-type Gauss map in pseudo-Euclidean space E4 2. Indian J. Pure Appl. Math. 46(1), 107-118 (2015).

Kim D.S., Kim J.R., Kim Y.H.: Cheng{Yau operator and Gauss map of surfaces of revolution. Bull. Malays. Math. Sci. Soc. 39(4), 1319-1327 (2016).

Kim Y.H., Turgay N.C.: Surfaces in E4 with L1-pointwise 1-type Gauss map. Bull. Korean Math. Soc. 50(3), 935-949 (2013).

Kühnel W.: Differential geometry. Curves-surfaces-manifolds. Third ed. Translated from the 2013 German ed. AMS, Providence, RI, 2015.

Levi-Civita T.: Famiglie di superficie isoparametriche nellordinario spacio euclideo. Rend. Acad. Lincei 26, 355-362 (1937).

Moore C.: Surfaces of rotation in a space of four dimensions. Ann. Math. 21, 81-93 (1919).

Moore C.: Rotation surfaces of constant curvature in space of four dimensions. Bull. Amer. Math. Soc. 26, 454-460 (1920).

Özkaldı S., Yaylı Y.: Tensor product surfaces in R4 and Lie groups. Bull. Malays. Math. Sci. Soc. (2)33 (1), 69-77 (2010).

Senoussi B., Bekkar M.: Helicoidal surfaces with ∆Jr = Ar in 3-dimensional Euclidean space. Stud. Univ. Babe¸s-Bolyai Math. 60(3), 437-448 (2015).

Stamatakis S., Zoubi H.: Surfaces of revolution satisfying ∆IIIx = Ax. J. Geom. Graph. 14(2), 181-186 (2010).

Takahashi T.: Minimal immersions of Riemannian manifolds. J. Math. Soc. Jpn. 18, 380-385 (1966).

Turgay N.C.: Some classifications of Lorentzian surfaces with finite type Gauss map in the Minkowski 4-space. J. Aust. Math. Soc. 99(3), 415-427 (2015).

Yoon D.W.: Some properties of the Clifford torus as rotation surfaces. Indian J. Pure Appl. Math. 34(6), 907-915 (2003).




DOI: https://doi.org/10.22190/FUMI211204063G

Refbacks

  • There are currently no refbacks.




© University of Niš | Created on November, 2013
ISSN 0352-9665 (Print)
ISSN 2406-047X (Online)