Research ArticleNo Access

Parabolic revolution surfaces of finite type in simply isotropic 3-spaces

Alev Kelleci Akbay

Department of Mathematics, Firat University, Elaziğ, 23 200, Turkey

https://doi.org/10.1142/S0219887821500341Cited by:0 (Source: Crossref)

Abstract

In this paper, we classify parabolic revolution surfaces in the three-dimensional simply isotropic space $𝕀^{3}$ under the condition

Δ J x i = λ i x i, J = I, II, <math display="block" altimg="eq-00002.gif"><mrow><msup><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>J</mi></mrow></msup><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo class="MathClass-rel">=</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo class="MathClass-punc">,</mo><mspace width="1em" class="quad"></mspace><mi>J</mi><mo class="MathClass-rel">=</mo><mstyle><mtext class="textrm" mathvariant="normal">I</mtext></mstyle><mo class="MathClass-punc">,</mo><mstyle><mtext class="textrm" mathvariant="normal">II</mtext></mstyle><mo class="MathClass-punc">,</mo></mrow></math>

where

$Δ^{J}$ is the Laplace operator with respect to first and second fundamental form and

$λ_{i}$ ,

$i = 1, 2, 3$ are some real numbers. Also, as an application, we give some explicit examples for these surfaces.

Keywords:

AMSC: 53A35, 53A40, 53A55

References

1. L. J. Alías, A. Ferrández and P. Lucas , Submanifolds in pseudo-Euclidean spaces satisfying the condition $Δ x = A x + b$ , Geom. Dedicata 42 (1992) 345–354. Crossref, Web of Science, Google Scholar
2. J. Arroyo, O. J. Garay and J. J. Mencía , On a family of surfaces of revolution of finite Chen-type, Kodai Math. J. 21 (1998) 73–80. Crossref, Google Scholar
3. M. E. Aydin and M. Ergut , Affine translation surfaces in the isotropic 3-space, Int. Electron. J. Geom. 10 (2017) 21–30. Web of Science, Google Scholar
4. M. Bekkar and H. Zoubir , Surfaces of revolution in the 3-dimensional Lorentz–Minkowski space satisfying $Δ x_{i} = λ_{i} x_{i}$ , Int. J. Contemp. Math. Sci. 3 (2008) 1173–1185. Google Scholar
5. B. Bukcu, M. K. Karacan and D. W. Yoon , Translation surfaces in the three-dimensional simply isotropic space $𝕀_{3}^{1}$ satisfying $Δ^{III} x_{i} = λ_{i} x_{i}$ , Konuralp J. Math. 4 (2016) 275–281. Google Scholar
6. B. Y. Chen , Total Mean Curvature and Submanifolds Finite Type (World Scientific, New Jersey, 1984). Link, Google Scholar
7. B. Y. Chen , Total Mean Curvature and Submanifolds of Finite Type, 2nd edn. (World Scientific, Hackensack, NJ, 2015). Google Scholar
8. B. Y. Chen , A report on submanifolds of finite type, Soochow J. Math. 22 (1996) 117–337. Google Scholar
9. B. Y. Chen , Surfaces of finite type in Euclidean 3-space, Bull. Soc. Math. Belg. Ser. B 39 (1987) 243–254. Google Scholar
10. B. Y. Chen, J. Morvan and T. Nore , Energy, tension and finite type maps, Kodai Math. J. 9 (1986) 406–418. Crossref, Google Scholar
11. L. C. B. da Silva , Differential geometry of invariant surfaces in simply isotropic and pseudo-isotropic spaces, Math. J. Okayama Univ. 63 (2021) 15–52. Google Scholar
12. F. Dillen, J. Pas and L. Verstraelen , On surfaces of finite type in Euclidean 3-space, Kodai Math. J. 13 (1990) 10–21. Crossref, Google Scholar
13. O. J. Garay , On a certain class of finite type surfaces of revolution, Kodai Math. J. 11 (1988) 25–31. Crossref, Google Scholar
14. O. J. Garay , An extension of Takahashi’s theorem, Geom. Dedicata 34 (1990) 105–112. Crossref, Web of Science, Google Scholar
15. C. B. Hamed and M. Bekkar , Helicoidal surfaces in the three-dimensional Lorentz–Minkowski space satisfying $Δ x_{i} = λ_{i} x_{i}$ , Int. J. Contemp. Math. Sci. 4 (2009) 311–327. Google Scholar
16. T. Hasanis and T. Vlachos , Coordinate finite-type submanifolds. Geom. Dedicata 37 (1991) 155–165. Crossref, Web of Science, Google Scholar
17. M. K. Karacan, D. W. Yoon and B. Bukcu , Translation surfaces in the three-dimensional simply isotropic space $𝕀_{3}^{1}$ , Int. J. Geom. Methods Modern Phys. 13 (2016) 1650088. Link, Web of Science, Google Scholar
18. M. K. Karacan, D. W. Yoon and B. Bukcu , Surfaces of revolution in the three-dimensional simply isotropic space $𝕀_{3}^{1}$ , Asia Pacific J. Math. 4 (2017) 1–10. Google Scholar
19. M. K. Karacan, D. W. Yoon and S. Kiziltug , Helicoidal surfaces in the three-dimensional simply isotropic space, $𝕀_{3}^{1}$ , Tamkang J. Math. 48 (2017) 123–134. Crossref, Web of Science, Google Scholar
20. M. K. Karacan, D. W. Yoon and N. Yuksel , Classification of some special types ruled surfaces in simply isotropic 3-space, An. Univ. Vest Timisoara Ser. Mat. — Inform. 55 (2017) 87–98. Google Scholar
21. H. Sachs , Isotrope Geometrie des Raumes (Vieweg, Braunschweig/Wiesbaden, 1990). Crossref, Google Scholar
22. B. Senoussi and M. Bekkar , Helicoidal surfaces with $Δ^{J} r = A r$ in 3-dimensional Euclidean space, Stud. Univ. Babeş-Bolyai Math. 60 (2015) 437–448. Web of Science, Google Scholar
23. T. Takahashi , Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan 18 (1966) 380–385. Crossref, Web of Science, Google Scholar
24. D. W. Yoon , Some classification of translation surfaces in Galilean 3-spaces, Int. J. Math. Anal. 6 (2012) 1355–1361. Google Scholar
25. D. W. Yoon , Surfaces of revolution in the three dimensional pseudo-Galilean space, Glas. Mat. 48 (2013) 415–428. Crossref, Web of Science, Google Scholar
26. A. Kelleci Akbay , Warped Translation Surfaces of Finite Type in Simply Isotropic 3-Spaces, Fundam. J. Math. Appl. 3(2) (2020) 137–143. Google Scholar
27. A. Kelleci, L. C. B. da Silva , Invariant surfaces with coordinate finite-type Gauss map in simply isotropic space, J. Math. Anal. Appl. 495(1) (2021) 124673. Crossref, Web of Science, Google Scholar