Abstract
In this paper, we investigate biharmonic submanifolds with parallel normalized mean curvature vector field in pseudo-Riemannian space forms and classify completely such pseudo-umbilical submanifolds. Also, we prove that such submanifolds have parallel mean curvature vector field under the assumption that they have diagonalizable shape operator with at most two distinct principal curvatures in the direction of the mean curvature vector field, and apply it to obtain a partial classification result.
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Arvanitoyeorgos, A., Defever, F., Kaimakamis, G., Papantoniou, V.J.: Biharmonic Lorentz hypersurfaces in \(\mathbb{E}^{4}_{1}\). Pac. J. Math. 229, 293–305 (2007)
Balmuş, A., Montaldo, S., Oniciuc, C.: Classification results for biharmonic submanifolds in spheres. Isr. J. Math. 168, 201–220 (2008)
Balmuş, A., Montaldo, S., Oniciuc, C.: Biharmonic PNMC submanifolds in spheres. Ark. Mat. 51, 197–221 (2013)
Caddeo, A., Montaldo, S., Oniciuc, C.: Biharmonic submanifolds of \(\mathbb{S}^3\). Int. J. Math. 12, 867–876 (2001)
Chen, B.-Y.: Surfaces with parallel normalized mean curvature vector. Monatsh. Math. 90, 185–194 (1980)
Chen, B.-Y.: Some open problems and conjectures on submanifolds of finite type. Soochow J. Math. 17, 169–188 (1991)
Chen, B.-Y.: Classification of marginally trapped Lorentzian flat surfaces in \(\mathbb{E}^4_2\) and its application to biharmonic surfaces. J. Math. Anal. Appl. 340, 861–875 (2008)
Chen, B.-Y.: Pseudo-Riemannian Geometry, \(\delta \)-Invariants and Applications. World Scientific, Hackensack (2011)
Chen, B.-Y., Ishikawa, S.: Biharmonic surfaces in pseudo-Euclidean spaces. Mem. Fac. Sci. Kyushu Univ. Ser. A 45, 323–347 (1991)
Chen, B.-Y., Ishikawa, S.: Biharmonic pseudo-Riemannian submanifolds in pseudo-Euclidean spaces. Kyushu. J. Math. 52, 167–185 (1998)
Defever, F., Kaimakamis, G., Papantoniou, V.: Biharmonic hypersurfaces of the 4-dimensional semi-Euclidean space \(\mathbb{E}^{4}_{s}\). J. Math. Anal. Appl. 315, 276–286 (2006)
Dimitrić, I.: Submanifolds of \(\mathbb{E}^{n+p}\) with harmonic mean curvature vector. Bull. Inst. Math. Acad. Sin. 20, 53–65 (1992)
Dong, Y.-X., Ou, Y.-L.: Biharmonic submanifolds of pseudo-Riemannian manifolds. J. Geom. Phys. 112, 252–262 (2017)
Eells, J., Lemarie, L.: Selected Topic in Harmonic Maps, CBMS 50. American Mathematical Society, Providence (1983)
Fu, Y.: Biharmonic hypersurfaces with three distinct principal curvatures in Euclidean space. Tohoku Math. J. 67, 465–479 (2015)
Fu, Y.: Biharmonic hypersurfaces with three distinct principal curvatures in spheres. Math. Nachr. 288, 763–774 (2015)
Gupta, R.S., Sharfuddin, A.: Biharmonic hypersurfaces in Euclidean space \(\mathbb{E}^5\). J. Geom. 107, 685–705 (2016)
Jiang, G.-Y.: 2-Harmonic maps and their first and second variational formulas. Chin. Ann. Math. Ser. A 7, 389–402 (1986)
Liu, J.-C., Du, L.: Classification of proper biharmonic hypersurfaces in pseudo-Riemannian space forms. Differ. Geom. Appl. 41, 110–122 (2015)
Liu, J.-C., Du, L., Zhang, J.: Minimality on biharmonic space-like submanifolds in pseudo-Riemannian space forms. J. Geom. Phys. 92, 69–77 (2015)
O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press, New York (1983)
Ou, Y.-L., Tang, L.: The generalized Chen’s conjecture on biharmonic submanifolds is false. Mich. Math. J. 61, 531–542 (2012)
Ouyang, C.-Z.: 2-Harmonic space-like submanifolds in pseudo-Riemannian space forms. Chin. Ann. Math. Ser. A 21, 649–654 (2000)
Sasahara, T.: Biharmonic submanifolds in nonflat Lorentz 3-space forms. Bull. Aust. Math. Soc. 85, 422–432 (2012)
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The authors are thankful to the referees for their many valuable suggestions and corrections that really improve the paper.
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Communicated by Young Jin Suh.
L. Du was supported by the Talent Engineering Funds (2012–2017) and the Major Project (No. TD2016ZD06). J. Zhang was supported by the Talent Engineering Funds (2013–2107) and the Ordinary Project (No. TD2016YB08).
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Du, L., Zhang, J. Biharmonic Submanifolds with Parallel Normalized Mean Curvature Vector Field in Pseudo-Riemannian Space Forms. Bull. Malays. Math. Sci. Soc. 42, 1469–1484 (2019). https://doi.org/10.1007/s40840-017-0556-y
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DOI: https://doi.org/10.1007/s40840-017-0556-y