## Taiwanese Journal of Mathematics

### HYPERSURFACES IN NON-FLAT PSEUDO-RIEMANNIAN SPACE FORMS SATISFYING A LINEAR CONDITION IN THE LINEARIZED OPERATOR OF A HIGHER ORDER MEAN CURVATURE

#### Abstract

We study hypersurfaces either in the pseudo-Riemannian De Sitter space $\mathbb{S}_t^{n+1} \subset \mathbb{R}_t^{n+2}$ orin the pseudo-Riemannian anti De Sitter space $\mathbb{H}_t^{n+1} \subset \mathbb{R}_{t+1}^{n+2}$ 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 $H_k$ is constant, the only hypersurfaces satisfying that condition are hypersurfaces with zero $(k+1)$-th mean curvature and constant $k$-th mean curvature, open pieces of a totally umbilical hypersurface in $\mathbb{S}^{n+1}_t$ ($\mathbb{S}^n_{t-1}(r)$, $r \gt 1$; $\mathbb{S}^n_t(r)$, $0 \lt r \lt 1$; $\mathbb{H}^n_{t-1}(-r)$, $r \gt 0$;  $\mathbb{R}^n_{t-1}$), open pieces of a totally umbilical hypersurface in $\mathbb{H}^{n+1}_t$ ($\mathbb{H}^n_t(-r)$, $r \gt 1$; $\mathbb{H}^n_{t-1}(-r)$, $0 \lt r \lt 1$; $\mathbb{S}^n_t(r)$, $r \gt 0$; $\mathbb{R}^n_t$), open pieces of a standard pseudo-Riemannian product in $\mathbb{S}_t^{n+1}$ ($\mathbb{S}_u^m(r) \times \mathbb{S}^{n-m}_v(\sqrt{1-r^2})$, $\mathbb{H}^m_{u-1}(-r) \times \mathbb{S}^{n-m}_v(\sqrt{1+r^2})$, $\mathbb{S}^m_u(r) \times \mathbb{H}^{n-m}_{v-1}(-\sqrt{r^2-1})$), open pieces of a standard pseudo-Riemannian product in $\mathbb{H}_t^{n+1}$ ($\mathbb{H}_u^m(-r) \times \mathbb{S}^{n-m}_v(\sqrt{r^2-1})$, $\mathbb{S}_u^m(r) \times \mathbb{H}^{n-m}_v(-\sqrt{1+r^2})$, $\mathbb{H}^m_u(-r) \times \mathbb{H}^{n-m}_{v-1}(-\sqrt{1-r^2})$) and open pieces of a quadratic hypersurface $\{x \in \mathbb{M}^{n+1}_t(c) \mid \langle Rx,x \rangle = d\}$, where $R$ is a self-adjoint constant matrix whose minimal polynomial is $\mu_R(z) = z^2 + az + b$, $a^2 - 4b \leq 0$, and $\mathbb{M}^{n+1}_t(c)$ stands for $\mathbb{S}_t^{n+1} \subset \mathbb{R}_t^{n+2}$ or  $\mathbb{H}_t^{n+1} \subset \mathbb{R}_{t+1}^{n+2}$.

#### Article information

Source
Taiwanese J. Math., Volume 17, Number 1 (2013), 15-45.

Dates
First available in Project Euclid: 10 July 2017

https://projecteuclid.org/euclid.twjm/1499705874

Digital Object Identifier
doi:10.11650/tjm.17.2013.1738

Mathematical Reviews number (MathSciNet)
MR3028856

Zentralblatt MATH identifier
1283.53067

#### Citation

Lucas, Pascual; Ramírez-Ospina, Héctor-Fabián. HYPERSURFACES IN NON-FLAT PSEUDO-RIEMANNIAN SPACE FORMS SATISFYING A LINEAR CONDITION IN THE LINEARIZED OPERATOR OF A HIGHER ORDER MEAN CURVATURE. Taiwanese J. Math. 17 (2013), no. 1, 15--45. doi:10.11650/tjm.17.2013.1738. https://projecteuclid.org/euclid.twjm/1499705874

#### References

• L. J. Al\ptmrs ías, 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\ptmrs ías, 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\ptmrs ías, 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\ptmrs ías 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\ptmrs ías and M. B. Kashani, Hypersurfaces in space forms satisfying the condition $L_k\psi=A\psi+b$, Taiwanese Journal of Mathematics, 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, 1959, New York.
• 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.
• H. Lawson, Local rigidity theorems for minimal hypersurfaces, Ann. Math., 89 (1969), 187-197.
• U. J. J. Leverrier, Sur les variations séculaire des élements des orbites pour les sept planétes principales, J. de Math., s.1, 5 (1840), 230ff.
• P. Lucas and H. F. Ram\ptmrs írez-Ospina, Hypersurfaces in the Lorentz-Minkowski space satisfying $L_k\psi=A\psi+b$, Geom. Dedicata, 153 (2011), 151-175.
• P. Lucas and H. F. Ram\ptmrs írez-Ospina, Hypersurfaces in non-flat Lorentzian space forms satisfying $L_k\psi=A\psi+b$, Taiwanese J. Math., 16 (2012), 1173-1203.
• P. Lucas and H. F. Ram\ptmrs írez-Ospina, Hypersurfaces in pseudo-Euclidean spaces satisfying the condition $L_k\psi=A\psi+b$, submitted for publication, 2011.
• 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.
• S. Roman, Advanced Linear Algebra, 3ed. Springer, 2008, New York.
• P. J. Ryan, Homogeneity and some curvature conditions for hypersurfaces, Tohoku Math. J., 21 (1969), 363-388.
• 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.