Home On η-biharmonic hypersurfaces in pseudo-Riemannian space forms
Article
Licensed
Unlicensed Requires Authentication

On η-biharmonic hypersurfaces in pseudo-Riemannian space forms

  • Li Du EMAIL logo and Jinjun Ren
Published/Copyright: October 16, 2022
Become an author with De Gruyter Brill

Abstract

In this paper, η-biharmonic hypersurfaces with constant scalar curvature in 5-dimensional pseudo-Riemannian space forms are studied. We prove that such hypersurfaces with diagonalizable shape operator have constant mean curvature, which gives an affirmative partial answer to the conjecture in [Arvanitoyeorgos, A.—Kaimakamis, F. G.: Hypersurfaces of type M23 in E24 with proper mean curvature vector, J. Geom. Phys. 63 (2013), 99–106]. As a result, we give several partial classification results.

MSC 2010: Primary 53C50
  1. (Communicated by Július Korbaš )

Acknowledgement

The authors would like to express their sincere gratitude to the referee for his/her valuable and detailed comments that help to improve the quality of the manuscript.

References

[1] Abe, N.—Koike, N.—Yamaguchi, N. S.: Congruence theorems for proper semi-Riemannian hypersurfaces in a real space form, Yokohama Math. J. 35 (1987), 123–136.Search in Google Scholar

[2] Arvanitoyeorgos, A.—Defever, F.—Kaimakamis, F. G.: Hypersurfaces of Es4 with proper mean curvature vector, J. Math. Soc. Japan 59 (2007), 797–809.10.2969/jmsj/05930797Search in Google Scholar

[3] Arvanitoyeorgos, A.—Kaimakamis, F. G.: Hypersurfaces of type M23 in E24 with proper mean curvature vector, J. Geom. Phys. 63 (2013), 99–106.10.1016/j.geomphys.2012.09.011Search in Google Scholar

[4] Arvanitoyeorgos, A.—Kaimakamis, F. G.—Magid, M.: Lorentz hypersurfaces in E14 satisfying ΔH⃗ = α H⃗, Illinois J. Math. 53 (2009), 581–590.10.1215/ijm/1266934794Search in Google Scholar

[5] Chen, B.Y.: Null two-type surfaces in 𝔼3 are circular cylinders, Kodai Math. J. 11 (1988), 295–299.10.2996/kmj/1138038880Search in Google Scholar

[6] Chen, B. Y.: Null two-type surfaces in Euclidean space, Proceedings of the symposium in honor of Cheng-Sung Hsu and Kung-Sing Shih: Algebra, Analysis, and Geometry (National Taiwan Univ. 1988), World Scientific, Publ. Teaneck, NJ, (1988), 1–18.Search in Google Scholar

[7] Chen, B. Y.: Report on submanifolds of finite type, Soochow J. Math. 22 (1996), 117–337.Search in Google Scholar

[8] Chen, B. Y.: Pseudo-Riemannian Geometry, δ-Invariants and Applications, World Scientific, Hackensack, NJ, 2011.10.1142/8003Search in Google Scholar

[9] Chen, B. Y.—Ishikawa, S.: Biharmonic surfaces in pseudo-Euclidean spaces, Mem. Fac. Sci. Kyushu Univ. A45 (1991), 323–347.10.2206/kyushumfs.45.323Search in Google Scholar

[10] Chen, B. Y.—Ishikawa, S.: Biharmonic pseudo-Riemannian submanifolds in pseudo-Euclidean spaces, Kyushu J. Math. 52 (1998), 167–185.10.2206/kyushujm.52.167Search in Google Scholar

[11] Defever, F.: Hypersurfaces of 𝔼4 satisfying Δ H⃗ = λH⃗, Michigan Math. J. 44 (1997), 355–363.10.1307/mmj/1029005710Search in Google Scholar

[12] Dong, Y. X.—Ou, Y. L.: Biharmonic submanifolds of pseudo-Riemannian manifolds, J. Geom. Phys. 112 (2017), 252–262.10.1016/j.geomphys.2016.11.019Search in Google Scholar

[13] Du, L.: Classification of η-biharmonic surfaces in non-flat Lorentz space forms, Mediterr. J. Math. 15 (2018), Art. No. 203.10.1007/s00009-018-1250-5Search in Google Scholar

[14] Du, L.—Zhang, J.: Biharmonic submanifolds with parallel normalized mean curvature vector field in pseudo-Riemannian space forms, Bull. Malays. Math. Sci. Soc. 42 (2019), 1469–1484.10.1007/s40840-017-0556-ySearch in Google Scholar

[15] Du, L.—Zhang, J.—Xie, X.: Hypersurfaces satisfying τ2(ϕ) = ητ(ϕ) in pseudo-Riemannian space forms, Math. Phys. Anal. Geom. 20 (2017), Art. No. 17.10.1007/s11040-017-9248-ySearch in Google Scholar

[16] Ferrández, A.—Lucas, P.: On surfaces in the 3-dimensional Lorentz–Minkowski space, Pacific J. Math. 152 (1992), 93–100.10.2140/pjm.1992.152.93Search in Google Scholar

[17] Ferrández, A.—Lucas, P.: Classifying hypersurfaces in the Lorentz–Minkowski space with a characteristic eigenvector, Tokyo J. Math. 15 (1992), 451–459.10.3836/tjm/1270129470Search in Google Scholar

[18] Fu, Y.: Biharmonic hypersurfaces with three distinct principal curvatures in Euclidean space, Tohoku Math. J. 67 (2015), 465–479.10.2748/tmj/1446818561Search in Google Scholar

[19] Fu, Y.: On biharmonic hypersurfaces with constant scalar curvatures in 𝕊5, Proc. Amer. Math. Soc. 143 (2015), 5399–5409.10.1090/proc/12677Search in Google Scholar

[20] Inoguchi, J.: Biminimal submanifolds in contact 3-manifolds, Balkan J. Geom. Appl. 12 (2007), 56–67.Search in Google Scholar

[21] Jiang, G. Y.: 2-harmonic isometric immersions between Riemannian manifolds, Chin. Ann. Math. 7A (1986), 130–144.Search in Google Scholar

[22] Jiang, G. Y.: 2-harmonic maps and their first and second variational formulas, Chin. Ann. Math. 7A (1986), 389–402.Search in Google Scholar

[23] Kengig, K.: Elementary Algebraic Geometry. GTM, vol. 44, Springer-Verlag, 1977.10.1007/978-1-4615-6899-5Search in Google Scholar

[24] KiliÇ, B.—Arslan, K.: Harmonic 1-type submanifolds of Euclidean spaces, Int. J. Math. Stat. 8A (2008), 47–53.Search in Google Scholar

[25] Liu, J. C.—Du, L.: Classification of proper biharmonic hypersurfaces in pseudo-Riemannian space forms, Diff. Geom. Appl. 41 (2015), 110–122.10.1016/j.difgeo.2015.05.001Search in Google Scholar

[26] Liu, J. C.—Yang, C.: Hypersurfaces in Esn+1 satisfying ΔH⃗ = λH⃗ with at most three distinct principal curvatures, J. Math. Anal. Appl. 419 (2014), 562–573.10.1016/j.jmaa.2014.04.066Search in Google Scholar

[27] Liu, J. C.—Yang, C.: Lorentz hypersurfaces in E1n+1 satisfying ΔH⃗ = λH⃗ with at most three distinct principal curvatures, J. Math. Anal. Appl. 434 (2016), 222–240.10.1016/j.jmaa.2015.09.017Search in Google Scholar

[28] Liu, J. C.—Yang, C.: Hypersurfaces in E1n+1 satisfying ΔH⃗ = λH⃗ with at most two distinct principal curvatures, J. Math. Anal. Appl. 451 (2017), 14–33.10.1016/j.jmaa.2017.01.090Search in Google Scholar

[29] O'Neill, B.: Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, 1983.Search in Google Scholar

[30] Ou, Y. L.: Some recent progress of biharmonic submanifolds, Contemp. Math. 674 (2016), 127–139.10.1090/conm/674/13559Search in Google Scholar

[31] Sasahara, T.: Biharmonic submanifolds in nonflat Lorentz 3-space forms, Bull. Aust. Math. Soc. 85 (2012), 422–432.10.1017/S0004972711002978Search in Google Scholar

Received: 2021-05-05
Accepted: 2021-09-05
Published Online: 2022-10-16
Published in Print: 2022-10-26

© 2022 Mathematical Institute Slovak Academy of Sciences

Downloaded on 14.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ms-2022-0086/html
Scroll to top button