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Minimals translation surfaces in a strict Walker 3-manifold

Year 2024, Volume: 73 Issue: 2, 554 - 568, 21.06.2024
https://doi.org/10.31801/cfsuasmas.1296991

Abstract

In this paper, we study minimal translations surfaces in a strict Walker 3-manifold. Based on the existence of two isometries, we classify minimal translation surfaces on this class of manifold. Some drawings have been added to illustrate the shape of certain surfaces.

References

  • Brozos-Vazquez, M., Garcia-Rio, E., Gilkey, P., Nikevic, S., Vazquez-Lorenzo, R., The Geometry of Walker Manifolds, Synthesis Lectures on Mathematics and Statistics, 5. Morgan and Claypool Publishers, Williston, VT, 2009.
  • Calvaruso, G., Van der Veken, J. Parallel surfaces in Lorentzian three-manifolds admitting a parallel null vector field, J. Phys. A: Math. Theor., 43 (2010) 325207, 9 pp. https://doi.org/10.1088/1751-8113/43/32/325207
  • Darboux, G. Lessons on the General Theory of Surfaces. I, II, Editions Jacques Gabay, Sceaux, 1993. 604 pp.
  • Diallo, A. S., Ndiaye, A., Niang, A., Minimal Graphs on Three-Dimensional Walker Manifolds, Proceedings of the First NLAGA-BIRS Symposium, Dakar, Senegal, 425-438, Trends Math., Birkhauser/Springer, Cham, 2020. https://doi.org/10.1007/978-3-030-57336-2-17
  • Dillen, F., Verstraele, L., Zafindratafa, G., A Generalization of the Translation Surfaces of Scherk, In: Differential Geometry in Honor of Radu Rosca. K. U. L., (1991), 107-109.
  • Inoguchi, I., Lopez, R., Munteanu, M. I., Minimal translation surfaces in the Heisenberg group Nil3, Geom. Dedicata, 16 (2012), 221-231. https://doi.org/10.1007/s10711-012-9702-8
  • Liu, H., Yu, Y., Affine translation surfaces in Euclidean 3-space, Proc. Japan Acad. Ser. A, 89 (2013), (9), 111-113. https://doi.org/10.3792/pjaa.89.111
  • Lopez, R., Minimal translation surfaces in hyperbolic space, Beitrage Alge. Geom., 52 (2011), 105-112. https://doi.org/10.1007/s13366-011-0008-z
  • Moruz, M., Munteanu, M. I., Minimal translation hypersurfaces in E4, J. Math. Analysis Appl., 439 (2016), 798-812. https://doi.org/10.1016/j.jmaa.2016.02.077
  • Niang, A., Ndiaye, A., Diallo, A. S., A Classification of Strict Walker 3-Manifold, Konuralp J. Math., 9(1) (2021), 148-153.
  • Scherk, H. F., Bemerkungen uber die kleinste Flache innerhalb gegebener Grenzen, J. Reine Angew. Math., 13 (1835), 185-208.
  • Yang, D., Fu, Y., On affine translation surfaces in affine space, J. Math. Analysis Appl., 440 (2016) 437-450. https://doi.org/10.1016/j.jmaa.2016.03.066
  • Yoon, D. W., Lee, C. W., Karacan, M. K., Some translation surfaces in the 3- dimensional Heisenberg group, Bull. Korean Math. Soc., 50(4) (2013), 1329-1343. https://doi.org/10.4134/BKMS.2013.50.4.1329
  • Yoon, D. W., Weighted minimal translation surfaces in the Galilean space with density, Open Math., 15 (1) (2017), 459-466. https://doi.org/10.1515/math-2017-0043
  • Walker, A. G., Canonical form for a Riemannian space with a parallel field of null planes, Quart. J. Math., Oxford, 1(1) (1950), 69-79. https://doi.org/10.1093/qmath/1.1.69
Year 2024, Volume: 73 Issue: 2, 554 - 568, 21.06.2024
https://doi.org/10.31801/cfsuasmas.1296991

Abstract

References

  • Brozos-Vazquez, M., Garcia-Rio, E., Gilkey, P., Nikevic, S., Vazquez-Lorenzo, R., The Geometry of Walker Manifolds, Synthesis Lectures on Mathematics and Statistics, 5. Morgan and Claypool Publishers, Williston, VT, 2009.
  • Calvaruso, G., Van der Veken, J. Parallel surfaces in Lorentzian three-manifolds admitting a parallel null vector field, J. Phys. A: Math. Theor., 43 (2010) 325207, 9 pp. https://doi.org/10.1088/1751-8113/43/32/325207
  • Darboux, G. Lessons on the General Theory of Surfaces. I, II, Editions Jacques Gabay, Sceaux, 1993. 604 pp.
  • Diallo, A. S., Ndiaye, A., Niang, A., Minimal Graphs on Three-Dimensional Walker Manifolds, Proceedings of the First NLAGA-BIRS Symposium, Dakar, Senegal, 425-438, Trends Math., Birkhauser/Springer, Cham, 2020. https://doi.org/10.1007/978-3-030-57336-2-17
  • Dillen, F., Verstraele, L., Zafindratafa, G., A Generalization of the Translation Surfaces of Scherk, In: Differential Geometry in Honor of Radu Rosca. K. U. L., (1991), 107-109.
  • Inoguchi, I., Lopez, R., Munteanu, M. I., Minimal translation surfaces in the Heisenberg group Nil3, Geom. Dedicata, 16 (2012), 221-231. https://doi.org/10.1007/s10711-012-9702-8
  • Liu, H., Yu, Y., Affine translation surfaces in Euclidean 3-space, Proc. Japan Acad. Ser. A, 89 (2013), (9), 111-113. https://doi.org/10.3792/pjaa.89.111
  • Lopez, R., Minimal translation surfaces in hyperbolic space, Beitrage Alge. Geom., 52 (2011), 105-112. https://doi.org/10.1007/s13366-011-0008-z
  • Moruz, M., Munteanu, M. I., Minimal translation hypersurfaces in E4, J. Math. Analysis Appl., 439 (2016), 798-812. https://doi.org/10.1016/j.jmaa.2016.02.077
  • Niang, A., Ndiaye, A., Diallo, A. S., A Classification of Strict Walker 3-Manifold, Konuralp J. Math., 9(1) (2021), 148-153.
  • Scherk, H. F., Bemerkungen uber die kleinste Flache innerhalb gegebener Grenzen, J. Reine Angew. Math., 13 (1835), 185-208.
  • Yang, D., Fu, Y., On affine translation surfaces in affine space, J. Math. Analysis Appl., 440 (2016) 437-450. https://doi.org/10.1016/j.jmaa.2016.03.066
  • Yoon, D. W., Lee, C. W., Karacan, M. K., Some translation surfaces in the 3- dimensional Heisenberg group, Bull. Korean Math. Soc., 50(4) (2013), 1329-1343. https://doi.org/10.4134/BKMS.2013.50.4.1329
  • Yoon, D. W., Weighted minimal translation surfaces in the Galilean space with density, Open Math., 15 (1) (2017), 459-466. https://doi.org/10.1515/math-2017-0043
  • Walker, A. G., Canonical form for a Riemannian space with a parallel field of null planes, Quart. J. Math., Oxford, 1(1) (1950), 69-79. https://doi.org/10.1093/qmath/1.1.69
There are 15 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Athoumane Nıang 0000-0003-4390-6438

Ameth Ndiaye 0000-0003-0055-1948

Abdoul Salam Diallo 0000-0003-4254-5829

Publication Date June 21, 2024
Submission Date May 14, 2023
Acceptance Date February 9, 2024
Published in Issue Year 2024 Volume: 73 Issue: 2

Cite

APA Nıang, A., Ndiaye, A., & Diallo, A. S. (2024). Minimals translation surfaces in a strict Walker 3-manifold. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 73(2), 554-568. https://doi.org/10.31801/cfsuasmas.1296991
AMA Nıang A, Ndiaye A, Diallo AS. Minimals translation surfaces in a strict Walker 3-manifold. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. June 2024;73(2):554-568. doi:10.31801/cfsuasmas.1296991
Chicago Nıang, Athoumane, Ameth Ndiaye, and Abdoul Salam Diallo. “Minimals Translation Surfaces in a Strict Walker 3-Manifold”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 73, no. 2 (June 2024): 554-68. https://doi.org/10.31801/cfsuasmas.1296991.
EndNote Nıang A, Ndiaye A, Diallo AS (June 1, 2024) Minimals translation surfaces in a strict Walker 3-manifold. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 73 2 554–568.
IEEE A. Nıang, A. Ndiaye, and A. S. Diallo, “Minimals translation surfaces in a strict Walker 3-manifold”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 73, no. 2, pp. 554–568, 2024, doi: 10.31801/cfsuasmas.1296991.
ISNAD Nıang, Athoumane et al. “Minimals Translation Surfaces in a Strict Walker 3-Manifold”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 73/2 (June 2024), 554-568. https://doi.org/10.31801/cfsuasmas.1296991.
JAMA Nıang A, Ndiaye A, Diallo AS. Minimals translation surfaces in a strict Walker 3-manifold. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2024;73:554–568.
MLA Nıang, Athoumane et al. “Minimals Translation Surfaces in a Strict Walker 3-Manifold”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 73, no. 2, 2024, pp. 554-68, doi:10.31801/cfsuasmas.1296991.
Vancouver Nıang A, Ndiaye A, Diallo AS. Minimals translation surfaces in a strict Walker 3-manifold. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2024;73(2):554-68.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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