Taiwanese Journal of Mathematics


Pascual Lucas and H. Fabian Ramirez-Ospina

Full-text: Open access


We study hypersurfaces either in the De Sitter space $\mathbb{S}_1^{n+1} \subset \mathbb{R}_1^{n+2}$ or in the anti De Sitter space $\mathbb{H}_1^{n+1} \subset \mathbb{R}_2^{n+2}$ whose position vector $\psi$ satisfies the condition $L_k \psi = A \psi + b$, where $L_k$ is the linearized operator of the $(k+1)$-th mean curvature of the hypersurface, for a fixed $k = 0,\dots,n-1$, $A$ is an $(n+2) \times (n+2)$ constant matrix and $b$ is a constant vector in the corresponding pseudo-Euclidean space. For every $k$, we prove that when $A$ is self-adjoint and $b = 0$, the only hypersurfaces satisfying that condition are hypersurfaces with zero $(k+1)$-th mean curvature and constant $k$-th mean curvature, open pieces of standard pseudo-Riemannian products in $\mathbb{S}_1^{n+1}$ ($\mathbb{S}_1^m(r) \times \mathbb{S}^{n-m}(\sqrt{1-r^2})$, $\mathbb{H}^m(-r) \times \mathbb{S}^{n-m}$ $(\sqrt{1+r^2})$, $\mathbb{S}_1^m(\sqrt{1-r^2}) \times \mathbb{S}^{n-m}(r)$, $\mathbb{H}^m(-\sqrt{r^2-1}) \times \mathbb{S}^{n-m}(r)$), open pieces of standard pseudo-Riemannian products in $\mathbb{H}_1^{n+1}$ ($\mathbb{H}_1^m(-r) \times \mathbb{S}^{n-m}(\sqrt{r^2-1})$, $\mathbb{H}^m(-\sqrt{1+r^2}) \times \mathbb{S}_1^{n-m}(r)$, $\mathbb{S}_1^m(\sqrt{r^2-1}) \times \mathbb{H}^{n-m}(-r)$, $\mathbb{H}^m(-\sqrt{1-r^2}) \times \mathbb{H}^{n-m}(-r)$) and open pieces of a quadratic hypersurface $\{x \in \mathbb{M}_{c}^{n+1} \;|\; = d\}$, where $R$ is a self-adjoint constant matrix whose minimal polynomial is $t^2+at+b$, $a^2-4b \leq 0$, and $\mathbb{M}_{c}^{n+1}$ stands for $\mathbb{S}_1^{n+1} \subset \mathbb{R}_1^{n+2}$ or $\mathbb{H}_1^{n+1} \subset \mathbb{R}_2^{n+2}$. When the $k$-th mean curvature is constant and $b$ is a non-zero constant vector, we show that the hypersurface is totally umbilical, and then we also obtain a classification result (see Theorem 2).

Article information

Taiwanese J. Math., Volume 16, Number 3 (2012), 1173-1203.

First available in Project Euclid: 18 July 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C50: Lorentz manifolds, manifolds with indefinite metrics 53B25: Local submanifolds [See also 53C40] 53B30: Lorentz metrics, indefinite metrics

linearized operator $L_k$ isoparametric hypersurface $k$-maximal hypersurface Takahashi theorem higher order mean curvatures Newton transformations


Lucas, Pascual; Ramirez-Ospina, H. Fabian. HYPERSURFACES IN NON-FLAT LORENTZIAN SPACE FORMS SATISFYING Lkψ = Aψ + b. Taiwanese J. Math. 16 (2012), no. 3, 1173--1203. doi:10.11650/twjm/1500406685. https://projecteuclid.org/euclid.twjm/1500406685

Export citation


  • L. J. Al\ipref s, A. Ferrández and P. Lucas, Surfaces in the 3-dimensional Lorentz-Minkowski space satisfying $\Delta x=Ax+B$, Pacific J. Math., 156(2) (1992), 201-208.
  • L. J. Al\ipref as, A. Ferrández and P. Lucas, Submanifolds in pseudo-Euclidean spaces satisfying the condition $\Delta x=Ax+B$, Geom. Dedicata, 42 (1992), 345-354.
  • L. J. Al\iprefl as, A. Ferrández and P. Lucas, Hypersurfaces in space forms satisfying the condition $\Delta x=Ax+B$, Trans. Amer. Math. Soc., 347 (1995), 1793-1801.
  • L. J. Al\iprefss as and N. Gürbüz, An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures, Geom. Dedicata, 121 (2006), 113-127.
  • L. J. Al\iprefs as and M. B. Kashani, Hypersurfaces in space forms satisfying the condition $L_k\psi=A\psi+b$, Taiwanese J. Math., 14 (2010), 1957-1978.
  • B.-Y. Chen and M. Petrovic, On spectral decomposition of immersions of finite type, Bull. Austral. Math. Soc., 44 (1991), 117-129.
  • S. Y. Cheng and S. T. Yau, Hypersurfaces with constant scalar curvature, Math. Ann., 225 (1977), 195-204.
  • F. Dillen, J. Pas and L. Verstraelen, On surfaces of finite type in Euclidean 3-space, Kodai Math. J., 13 (1990), 10-21.
  • V. N. Faddeeva, Computational Methods of Linear Algebra, Dover Publ. Inc, New York, 1959.
  • O. J. Garay, An extension of Takahashi's theorem, Geom. Dedicata, 34 (1990), 105-112.
  • J. Hahn, Isoparametric hypersurfaces in the pseudo-Riemannian space forms, Math. Z., 187 (1984), 195-208.
  • T. Hasanis and T. Vlachos, Hypersurfaces of $E^{n+1}$ satisfying $\Delta x = Ax + B$, J. Austral. Math. Soc. Ser. A, 53 (1992), 377-384.
  • U. J. J. Leverrier, Sur les variations séculaire des \'lements des orbites pour les sept planétes principales, J. de Math., s.1, 5 (1840), 230ff.
  • P. Lucas and H. F. Ram\iprefll rez-Ospina, Hypersurfaces in the Lorentz-Minkowski space satisfying $L_k\psi=A\psi+b$, Geom. Dedicata, 153 (2011), 151-175.
  • M. A. Magid, Lorentzian isoparametric hypersurfaces, Pacific J. Math., 118 (1985), 165-197.
  • K. Nomizu, On isoparametric hypersurfaces in the Lorentzian space forms, Japan J. Math. $($N.S.$)$, 7 (1981), 217-226.
  • B. O'Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, London, 1983.
  • R. Reilly, Variational properties of functions of the mean curvatures for hypersurfaces in space forms, J. Diff. Geom., 8 (1973), 465-477.
  • T. Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan, 18 (1966), 380-385.
  • L. Xiao, Lorentzian isoparametric hypersurfaces in $\H_1^{n+1}$, Pacific J. Math., 189 (1999), 377-397.
  • L. Zhen-Qi and X. Xian-Hua, Space-like Isoparametric Hypersurfaces in Lorentzian Space Forms, J. Nanchang Univ. Nat. Sci. Ed., 28 (2004), 113-117.