Taiwanese Journal of Mathematics


Pascual Lucas and Héctor-Fabián Ramírez-Ospina

Full-text: Open access


In this article, we study $L_k$-finite-type hypersurfaces $M^n$ of a hyperbolic space $\mathbb{H}^{n+1}\subset\mathbb{R}^{n+2}_1$, for $k\geq 1$. In the 3-dimensional case, we obtain the following classification result. Let $\psi:M^3\rightarrow\mathbb{H}^{4}\subset\mathbb{R}^5_1$ be an orientable hypersurface with constant $k$-th mean curvature $H_k$, which is not totally umbilical. Then $M^3$ is of $L_k$-2-type if and only if $M^3$ is an open portion of a standard Riemannian product $\mathbb{H}^1(r_1)\times\mathbb{S}^{2}(r_2)$ or $\mathbb{H}^2(r_1)\times\mathbb{S}^{1}(r_2)$, with $-r_1^2+r_2^2=-1$. In the $n$-dimensional case, we show that a hypersurface $M^n\subset\mathbb{H}^{n+1}$, with constant $k$-th mean curvature $H_k$ and having at most two distinct principal curvatures, is of $L_k$-2-type if and only if $M^n$ is an open portion of a Riemannian product $\mathbb{H}^m(r_1)\times\mathbb{S}^{n-m}(r_2)$, with $-r_1^2+r_2^2=-1$, for some integer $m\in\{1,\dots,n-1\}$. In the case $k=n-1$ we drop the condition on the principal curvatures of the hypersurface $M^n$, and prove that if $M^n\subset\mathbb{H}^{n+1}$ is an orientable $H_{n-1}$-hypersurface of $L_{n-1}$-2-type then its Gauss-Kronecker curvature $H_{n}$ is a nonzero constant.

Article information

Taiwanese J. Math., Volume 19, Number 1 (2015), 221-242.

First available in Project Euclid: 4 July 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C40: Global submanifolds [See also 53B25] 53B25: Local submanifolds [See also 53C40]

hyperbolic hypersurface linearized operator $L_k$ $L_k$-finite-type hypersurface higher order mean curvatures Newton transformations


Lucas, Pascual; Ramírez-Ospina, Héctor-Fabián. $L_k$-2-TYPE HYPERSURFACES IN HYPERBOLIC SPACES. Taiwanese J. Math. 19 (2015), no. 1, 221--242. doi:10.11650/tjm.19.2015.4603. https://projecteuclid.org/euclid.twjm/1499133627

Export citation


  • 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.
  • A. L. Besse, Einstein Manifolds, Ergeb. Math. Grenzgeb. (3), 10. Springer-Verlag, Berlin, 1987.
  • E. Cartan, Familles de surfaces isoparamétriques dans les espaces a courbure constante, Ann. Mat. Pura Appl., 17 (1938), 177-191.
  • E. Cartan, Sur des familles remarquables d'hypersurfaces isoparamétriques dans les espaces sphériques, Math. Z., 45 (1939), 335-367.
  • E. Cartan, Sur Quelque Familles Remarquables d'hypersurfaces, C. R. Congrès Math. Liège, 1939, 30-41.
  • E. Cartan, Sur des familles d'hypersurfaces isoparamétriques des espaces sphériques à 5 et à 9 dimensions, Univ. Nac. Tucumán Rev. Ser. A, 1 (1940), 5-22.
  • B. Y. Chen, Total Mean Curvature and Submanifolds of Finite Type, Series in Pure Mathematics, 1. World Scientific Publishing Co., Singapore, 1984.
  • B. Y. Chen, Finite Type Submanifolds and Generalizations, University of Rome, Rome, 1985.
  • B. Y. Chen, Finite type submanifolds in pseudo-Euclidean spaces and applications, Kodai Math. J., 8 (1985), 358-375.
  • B. Y. Chen, Finite-type pseudo-Riemannian submanifolds, Tamkang J. Math., 17(2) (1986), 137-151.
  • B. Y. Chen, 2-type submanifolds and their applications, Chinese J. Math., 14 (1986), 1-14.
  • B. Y. Chen, M. Barros and O. J. Garay, Spherical finite type hypersurfaces, Algebras Groups Geom., 4 (1987), 58-72.
  • B. Y. Chen, Submanifolds of finite type in hyperbolic spaces, Chinese J. Math., 20(1) (1992), 5-21.
  • B. Y. Chen, A report on submanifolds of finite type, Soochow J. Math., 22(2) (1996), 117-337.
  • B. Y. Chen, Some open problems and conjectures on submanifolds of finite type: recent development, Tamkang J. Math., 45(1) (2014), 87-108.
  • T. Hasanis and T. Vlachos, Spherical 2-type hypersurfaces, J. Geom., 40 (1991), 82-94.
  • S. M. B. Kashani, On some $L_1$-finite type (hyper)surfaces in $\R^{n+1}$, Bull. Korean Math. Soc., 46 (2009), 35-43.
  • S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. I, Wiley-Interscience, New York, NY, USA, 1963; Vol. II, 1969.
  • P. Lucas and H. F. Ram\ptmrs írez-Ospina, Hypersurfaces in pseudo-Euclidean spaces satisfying a linear condition on the linearized operator of a higher order mean curvature, Differential Geom. Appl., 31 (2013), 175-189.
  • P. Lucas and H. F. Ram\ptmrs írez-Ospina, $L_k$-2-type Hypersurfaces in $\S^4$, submitted for publication.
  • A. Mohammadpouri and S. M. B. Kashani, On some $L_k$-finite-type Euclidean hypersurfaces, ISRN Geom., Vol. 2012, article ID 591296, 23 pages.
  • B. O'Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, 1983, New York, London.
  • T. Otsuki, Minimal hypersurfaces in a Riemannian manifold of constant curvature, Amer. J. Math., 92(1) (1970), 145-173.
  • R. Reilly, Variational properties of functions of the mean curvatures for hypersurfaces in space forms, J. Differential Geom., 8 (1973), 465-477.
  • B. Segre, Famiglie di ipersuperficie isoparametrische negli spazi euclidei ad un qualunque numero di demesioni, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 27 (1938), 203-207.
  • C. Somigliana, Sulle relazione fra il principio di Huygens e l'ottica geometrica, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 54 (1918-1919), 974-979.