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
(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
[3] Arvanitoyeorgos, A.—Kaimakamis, F. G.: Hypersurfaces of type
[4] Arvanitoyeorgos, A.—Kaimakamis, F. G.—Magid, M.: Lorentz hypersurfaces in
[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
[27] Liu, J. C.—Yang, C.: Lorentz hypersurfaces in
[28] Liu, J. C.—Yang, C.: Hypersurfaces in
[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
© 2022 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Regular Papers
- Generalized hyperharmonic number sums with reciprocal binomial coefficients
- Gini index on generalized r-partitions
- Multiplicative functions of special type on Piatetski-Shapiro sequences
- Strengthenings of Young-type inequalities and the arithmetic geometric mean inequality
- Generalizations of the steffensen integral inequality for pseudo-integrals
- Subordination-implication problems concerning the nephroid starlikeness of analytic functions
- Boundedness and almost periodicity of solutions of linear differential systems
- On variational approaches for fractional differential equations
- Approximity of asymmetric metric spaces
- Approximation theorems for the new construction of Balázs operators and its applications
- On η-biharmonic hypersurfaces in pseudo-Riemannian space forms
- Chen’s first inequality for hemi-slant warped products in nearly trans-Sasakian manifolds
- Induced mappings on symmetric products of Hausdorff spaces
- The Teissier-G family of distributions: Properties and applications
- A new extension of the beta generator of distributions
- A new family of compound exponentiated logarithmic distributions with applications to lifetime data
- On two correlated linear models with common and different parameters
- On some applications of Duhamel operators
Articles in the same Issue
- Regular Papers
- Generalized hyperharmonic number sums with reciprocal binomial coefficients
- Gini index on generalized r-partitions
- Multiplicative functions of special type on Piatetski-Shapiro sequences
- Strengthenings of Young-type inequalities and the arithmetic geometric mean inequality
- Generalizations of the steffensen integral inequality for pseudo-integrals
- Subordination-implication problems concerning the nephroid starlikeness of analytic functions
- Boundedness and almost periodicity of solutions of linear differential systems
- On variational approaches for fractional differential equations
- Approximity of asymmetric metric spaces
- Approximation theorems for the new construction of Balázs operators and its applications
- On η-biharmonic hypersurfaces in pseudo-Riemannian space forms
- Chen’s first inequality for hemi-slant warped products in nearly trans-Sasakian manifolds
- Induced mappings on symmetric products of Hausdorff spaces
- The Teissier-G family of distributions: Properties and applications
- A new extension of the beta generator of distributions
- A new family of compound exponentiated logarithmic distributions with applications to lifetime data
- On two correlated linear models with common and different parameters
- On some applications of Duhamel operators