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## AFFINE TRANSLATION SURFACES IN 3-DIMENSIONAL EUCLIDEAN SPACE SATISFYING $\Delta r_{i}=\lambda _{i}r_{i}$

#### BENDEHIBA SENOUSSI [1] , MOHAMMED BEKKAR [2]

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In this paper we study the affine translation surfaces in 3-dimensional Euclidean space $\mathbb{E}^{3}$ under the condition $\Delta r_{i}=\lambda _{i}r_{i}$, where $\lambda _{i}\in \mathbb{R}$ and $\Delta$ denotes the Laplace operator. We obtain the complete classification for those ones.
Ane translation surfaces, nite type immersion, Laplacian operator
• [1] M. Bekkar and B. Senoussi, Factorable surfaces in the three-dimensional Euclidean and Lorentzian spaces satisfying $\Delta r_{i}=\lambda _{i}r_{i},$ J. Geom. 103 (2012), 17 - 29.
• [2] M. Bekkar and B. Senoussi, Translation surfaces in the 3-dimensional space satisfying $\Delta ^{III}r_{i}=\mu _{i}r_{i},$ J. Geom. 103 (2012), 367-374.
• [3] Chr. Beneki, G. Kaimakamis and B.J. Papantoniou, Helicoidal surfaces in the three dimensional Minkowski space, J. Math. Appl. 275 (2002), 586-614.
• [4] B.-Y. Chen, Total mean curvature and submanifolds of nite type, World Scienti c, Singapore. (1984).
• [5] M. Choi and Y.H. Kim, Characterization of the helicoid as ruled surfaces with pointwise 1-type Gauss map, Bull. Korean Math. Soc. 38 (2001), 753-761.
• [6] M. Choi, Y.H. Kim, H. Liu and D.W. Yoon, Helicoidal surfaces and their Gauss map in Minkowski 3-Space, Bull. Korean Math. Soc. 47 (2010), 859-881.
• [7] F. Dillen, J. Pas and L. Verstraelen, On surfaces of nite type in Euclidean 3-space, Kodai Math. J. 13 (1990), 10-21.
• [8] A. Ferrandez, O.J. Garay and P. Lucas, On a certain class of conformally at Euclidean hypersurfaces, Proc. of the Conf, in Global Analysis and Global Differential Geometry, Berlin. (1990).
• [9] G. Kaimakamis, B.J. Papantoniou and K. Petoumenos, Surfaces of revolution in the 3-dimensional Lorentz-Minkowski space $\mathbb{E}_{1}^{3}$ satisfying $\Delta ^{III}\overrightarrow{r}=A\overrightarrow{r}$, Bull. Greek. Math. Soc. 50 (2005), 76-90.
• [10] H. Liu, Translation surfaces with constant mean curvature in 3-dimensional spaces, J. Geom. 64 (1999), 141-149.
• [11] H. Liu and Y. Yu, Ane translation surfaces in Euclidean 3 -space, Proc. Japan Acad. 89 (2013), 111-113.
• [12] R. Lopez, Minimal translation surfaces in hyperbolic space, Beitr. Algebra Geom. 52 (2011), 105-112.
• [13] R. Lopez and M. I. Munteanu, Minimal translation surfaces in Sol3, J. Math. Soc. Japan, 64 (2012), 985-1003.
• [14] M. I. Munteanu and A. I. Nistor, Polynomial translationWeingarten surfaces in 3-dimensional Euclidean space, Proceedings of the VIII International Colloquium on Di erential Geometry, World Scienti c. (2009), 316-320.
• [15] B. Senoussi and M. Bekkar, Helicoidal surfaces in the 3-dimensional Lorentz - Minkowski space $\mathbb{E}_{1}^{3}$ satisfying $\Delta ^{III}r=Ar$, Tsukuba J. Math. 37 (2013), 339 - 353.
• [16] K. Seo, Translation hypersurfaces with constant curvature in space forms, Osaka J. Math. 50 (2013), 631-641.
• [17] S. Stamatakis and H. Al-Zoubi, Surfaces of revolution satisfying $\Delta ^{III}x=Ax$, J. Geom. Graph. 14 (2010), 181-186.
• [18] B. O'Neill, Semi-Riemannian geometry with applications to relativity, Academic Press, Waltham. (1983).
• [19] T. Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan. 18 (1966), 380-385.
• [20] L. Verstraelen, J. Walrave, S. Yaprak, The minimal translation surfaces in Euclidean space, Soochow J. Math. 20 (1994), 77-82.
Konular Mühendislik ve Temel Bilimler Articles Yazar: BENDEHIBA SENOUSSIÜlke: Algeria Yazar: MOHAMMED BEKKARÜlke: Algeria
 Bibtex @araştırma makalesi { konuralpjournalmath343312, journal = {Konuralp Journal of Mathematics}, issn = {}, eissn = {2147-625X}, address = {Mehmet Zeki SARIKAYA}, year = {2017}, volume = {5}, pages = {47 - 53}, doi = {}, title = {AFFINE TRANSLATION SURFACES IN 3-DIMENSIONAL EUCLIDEAN SPACE SATISFYING \$\\Delta r\_\{i\}=\\lambda \_\{i\}r\_\{i\}\$}, key = {cite}, author = {BEKKAR, MOHAMMED and SENOUSSI, BENDEHIBA} } APA SENOUSSI, B , BEKKAR, M . (2017). AFFINE TRANSLATION SURFACES IN 3-DIMENSIONAL EUCLIDEAN SPACE SATISFYING $\Delta r_{i}=\lambda _{i}r_{i}$. Konuralp Journal of Mathematics, 5 (2), 47-53. Retrieved from http://dergipark.gov.tr/konuralpjournalmath/issue/28490/343312 MLA SENOUSSI, B , BEKKAR, M . "AFFINE TRANSLATION SURFACES IN 3-DIMENSIONAL EUCLIDEAN SPACE SATISFYING $\Delta r_{i}=\lambda _{i}r_{i}$". Konuralp Journal of Mathematics 5 (2017): 47-53 Chicago SENOUSSI, B , BEKKAR, M . "AFFINE TRANSLATION SURFACES IN 3-DIMENSIONAL EUCLIDEAN SPACE SATISFYING $\Delta r_{i}=\lambda _{i}r_{i}$". Konuralp Journal of Mathematics 5 (2017): 47-53 RIS TY - JOUR T1 - AFFINE TRANSLATION SURFACES IN 3-DIMENSIONAL EUCLIDEAN SPACE SATISFYING $\Delta r_{i}=\lambda _{i}r_{i}$ AU - BENDEHIBA SENOUSSI , MOHAMMED BEKKAR Y1 - 2017 PY - 2017 N1 - DO - T2 - Konuralp Journal of Mathematics JF - Journal JO - JOR SP - 47 EP - 53 VL - 5 IS - 2 SN - -2147-625X M3 - UR - Y2 - 2017 ER - EndNote %0 Konuralp Journal of Mathematics AFFINE TRANSLATION SURFACES IN 3-DIMENSIONAL EUCLIDEAN SPACE SATISFYING $\Delta r_{i}=\lambda _{i}r_{i}$ %A BENDEHIBA SENOUSSI , MOHAMMED BEKKAR %T AFFINE TRANSLATION SURFACES IN 3-DIMENSIONAL EUCLIDEAN SPACE SATISFYING $\Delta r_{i}=\lambda _{i}r_{i}$ %D 2017 %J Konuralp Journal of Mathematics %P -2147-625X %V 5 %N 2 %R %U ISNAD SENOUSSI, BENDEHIBA , BEKKAR, MOHAMMED . "AFFINE TRANSLATION SURFACES IN 3-DIMENSIONAL EUCLIDEAN SPACE SATISFYING $\Delta r_{i}=\lambda _{i}r_{i}$". Konuralp Journal of Mathematics 5 / 2 (Ekim 2017): 47-53.