On hypersurfaces of Lorentzian standard 4-space forms satisfying a biconservativity condition

Document Type : Research Paper

Author

Department of Mathematics, Faculty of Basic Sciences, University of Maragheh, Maragheh, Iran

Abstract

In this manuscript, we consider an extended version of biconservativity condition (namely, ${\textrm C}$-biconservativity) on the Riemannian hypersurfaces of Lorentzian standard 4-space forms. This new condition is obtained by substituting the Cheng-Yau operator ${\textrm C}$ instead of the Laplace operator $\Delta$. We show that every ${\textrm C}$-biconservative Riemannian hypersurface of a Lorentzian 4-space form with constant mean curvature has constant scalar curvature.

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References

[1] Alias, L. J., Ferr´andez, A. & Lucas, P. (1995). Hypersurfaces in space forms satisfying the condition Δx = Ax + B. Trans. Amer. Math. Soc., 347(5), 1793-1801. http://doi.org/10.1090/S0002-9947-1995-1257095-8
[2] Alias, L. J. & G¨urb¨uz, N. (2006). An extension of Takahashi theorem for the linearized * operators of the higher order mean curvatures. Geom. Ded., 121, 113-127. http://doi.org/10.1007/s10711-006-9093-9
[3] Alias, L. J. & Kashani, S. M. B. (2010). Hypersurfaces in space forms satisfying the condition Lkx = Ax + b. Taiwanese J. of Math., 14(5), 1957-1977. https://doi.org/10.48550/arXiv.0908.3595
[4] Caddeo, R., Montaldo, S., Oniciuc, C. & Piu, P. (2014). Surfaces in three-dimensional space forms with divergence-free stress-bienergy tensor. Ann. Mat. Pura Appl., 193(2), 529￿550. http://doi.org/10.1007/s10231-012-0289-3
[5] Chen, B. Y. (1991). Some open problems and conjectures on submanifolds of finite type. Soochow J. Math., 17, 169￿188.
[6] Cheng, S. Y. & Yau, S. T. (1977). Hypersurfaces with constant scalar curvature. Math. Ann., 225(3), 195-204.  https://link.springer.com/article/10.1007/BF01425237
[7] Do Carmo, M. & Dajczer, M. (1983). Rotation Hypersurfaces in Spaces of Constant Curvature. Trans. Amer. Math. Soc., 277(2), 685￿709. https://link.springer.com/chapter/10.1007/978-3-642-25588-5−17
[8] Eells, J. & Sampson, J. (1964). Harmonic mappings of Riemannian manifolds. Am. J. Math., 86, 109￿160. http://doi.org/10.1142/9789814360197−0001
[9] Fetcu, D. & Oniciuc, C. (2022). Biharmonic and biconservative hypersurfaces in space forms. Diff. Geom. and Global anal. in honor of Tadashi Nagano, 65￿90. Contemp. Math., 777, Amer. Math. Soc., (Providence), RI. https://doi.org/10.48550/arXiv.2012.12476
[10] Fetcu, D., Oniciuc, C. & Pinheiro, A. L. (2015). CMC biconservative surfaces in Sn ×R and Hn × R. J. Math. Anal. Appl., 425, 588-609. doi.org/10.1016/j.jmaa.2014.12.051.
[11] Fu, Y. & Turgay, N. C. (2016). Complete classification of biconservative hypersurfaces with diagonalizable shape operator in Minkowaski 4-space. Inter. J. of Math., 27(5), 1650041. https://doi.org/10.1142/S0129167X16500415.
[12] Gupta, R. S. (2019). Biconservative hypersurfaces in Euclidean 5-space. Bull. Iran. Math. Soc., 45(4), 1117￿1133. https://link.springer.com/journal/41980/volumes-andissues/45-4.
[13] Hasanis, T. & Vlachos, T. (1995). Hypersurfaces in E4 with harmonic mean curvature vector field. Math. Nachr., 172, 145-169. https://doi.org/10.1002/mana.19951720112.
[14] Jiang, G. Y. (1987). The conservation law for 2-harmonic maps between Riemannian manifolds. Acta Math. Sin., 30, 220￿225.
[15] Kashani, S. M. B. (2009). On some L1-finite type (hyper)surfaces in Rn+1. Bull. Korean Math. Soc., 46(1), 35-43. http://doi.org/10.4134/BKMS.2009.46.1.035.
[16] Nomizu, K. & Smyth, B. (1969). A formula of Simons type and hypersurfaces with constant mean curvature. J. Diff. Geom., 3, 367￿377. http://doi.org/10.4310/jdg/1214429059.
[17] O’Neill, B. (1983). Semi-Riemannian Geometry with Applicatins to Relativity (2nd ed.), Acad. Press Inc.
[18] Petrov, A. Z. (1969). Einstein Spaces, Pergamon Press, Hungary, Oxford and New York.
[19] Pashaie, F. & Kashani, S. M. B. (2013). Spacelike hypersurfaces in Riemannian or Lorentzian space forms satisfying Lkx = Ax+b. Bull. Iran. Math. Soc., 39(1), 195-213. http://bims.iranjournals.ir/article−338−564a4b1f52aa0e44c763a92cfc8189b5.pdf.
[20] Pashaie, F. & Kashani, S. M. B. (2014). Timelike hypersurfaces in the Lorentzian standard space forms satisfying Lkx = Ax + b. Mediterr. J. Math., 11(2), 755-773. http://doi.org/10.1007/s00009-013-0336-3.
[21] Pashaie, F. (2023). Some L1-biconservative Lorentzian hypersurfaces in the Lorentz-Minkowski spaces. Kragujevac J. of Math., 47(2), 229-243. http://doi.org/10.46793/KgJMat2302.229P.
[22] Pashaie, F. (2023). On biconservative hypersurfaces in the Lorentz-Minkowski 4-space. Sahand Comm. in Math. Analysis, 20(3), 1-18. https://doi.org/10.22130/scma.2023.1982815.1215.
[23] Pashaie, F. (2023). Biconservative Lorentz hypersurfaces with at least three principal curvatures. Tamkang J. of Math., 54, (2023). http://doi.org/10.5556/j.tkjm.54.2023.4876.
[24] Reilly, R. C. (1973). Variational properties of functions of the mean curvatures for hypersurfaces in space forms. J.Differential Geom., 8(3), 465-477. http://doi.org/10.4310/jdg/1214431802.
[25] Turgay, N. C. (2015). H-hypersurfaces with 3 distinct principal curvatures in the Euclidean spaces. Ann. di Mat. Pura Appl., 194(6), 1795￿1807. http://doi.org/10.1007/s10231-014-0445-z.
[26] Turgay, N. C. & Upadhyay, A. (2019). On biconservative hypersurfaces in 4-dimensional Riemannian space forms. Math. Nachr., 292(4), 905￿921. https://doi.org/10.48550/arXiv.1702.05469.
[27] Upadhyay, A. & Turgay, N. C. (2016). A Classification of Biconservative Hypersurfaces in a Pseudo-Euclidean Space. J. Math. Anal. Appl., 444, 1703￿1720. https://doi.org/10.1016/j.jmaa.2016.07.053.
[28] Wei, G. (2008). Complete hypersurfaces in a Euclidean space Rn+1 with constant mth mean curvature. Diff. Geom. Appl., 26, 298-306. https://doi.org/10.1016/j.difgeo.2007.11.021.
[29] Zhen-Qi, L. & Xian-Hua, X. (2006). Space-like isoparametric hypersurfaces in Lorentzian space forms. Front. Math. China, 1(1), 130-137. https://doi.org/10.1007/s11464-005-0026-y
 
Journal of Mahani Mathematical Research
Volume 13, Issue 1 - Serial Number 26
November 2023
Pages 333-346

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History

  • Receive Date: 24 January 2023
  • Revise Date: 31 July 2023
  • Accept Date: 14 September 2023

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