Geometry/Differential geometry
On standard imbeddings of hyperbolic spaces in the Minkowski space

Présenté par : the Editorial Board

Kim, Dong-Soo1 ; Kim, Young Ho2 ; Yoon, Dae Won3
1 Department of Mathematics, Chonnam National University, Gwangju 500-757, South Korea
2 Department of Mathematics, Kyungpook National University, Daegu 702-701, South Korea
3 Department of Mathematics Education and RINS, Gyeongsang National University, Jinju 660-701, South Korea
Comptes Rendus. Mathématique, Tome 352 (2014) no. 12, pp. 1033-1038.

Nous établissons quelques caractérisations des plongements standard d'espaces hyperboliques dans l'espace de Minkowski Ln+1 de dimension n+1, avec des propriétés intrinsèques et extrinsèques comme la surface n-dimensionnelle des sections coupées par des hyperplans, le volume en n+1 dimensions de régions entre des hyperplans parallèles et la surface n-dimensionnelle de régions entre des hyperplans parallèles. De la même façon, nous donnons une réponse affirmative partielle à la question A suggérée dans [6], qui concerne la caractérisation d'hypersphères dans l'espace Euclidien En+1 de dimension n+1.

We establish some characterizations of the standard imbeddings of hyperbolic spaces in the (n+1)-dimensional Minkowski space Ln+1 with intrinsic and extrinsic properties such as the n-dimensional area of the sections cut off by hyperplanes, the (n+1)-dimensional volume of regions between parallel hyperplanes, and the n-dimensional surface area of regions between parallel hyperplanes. In the same manner, we give an affirmatively partial answer to Question A suggested in [6], which is for the characterization of hyperspheres in the (n+1)-dimensional Euclidean space En+1.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2014.09.003
Kim, Dong-Soo 1 ; Kim, Young Ho 2 ; Yoon, Dae Won 3

1 Department of Mathematics, Chonnam National University, Gwangju 500-757, South Korea
2 Department of Mathematics, Kyungpook National University, Daegu 702-701, South Korea
3 Department of Mathematics Education and RINS, Gyeongsang National University, Jinju 660-701, South Korea 
@article{CRMATH_2014__352_12_1033_0,
     author = {Kim, Dong-Soo and Kim, Young Ho and Yoon, Dae Won},
     title = {On standard imbeddings of hyperbolic spaces in the {Minkowski} space},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1033--1038},
     publisher = {Elsevier},
     volume = {352},
     number = {12},
     year = {2014},
     doi = {10.1016/j.crma.2014.09.003},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.crma.2014.09.003/}
}
TY  - JOUR
AU  - Kim, Dong-Soo
AU  - Kim, Young Ho
AU  - Yoon, Dae Won
TI  - On standard imbeddings of hyperbolic spaces in the Minkowski space
JO  - Comptes Rendus. Mathématique
PY  - 2014
SP  - 1033
EP  - 1038
VL  - 352
IS  - 12
PB  - Elsevier
UR  - https://www.numdam.org/articles/10.1016/j.crma.2014.09.003/
DO  - 10.1016/j.crma.2014.09.003
LA  - en
ID  - CRMATH_2014__352_12_1033_0
ER  - 
%0 Journal Article
%A Kim, Dong-Soo
%A Kim, Young Ho
%A Yoon, Dae Won
%T On standard imbeddings of hyperbolic spaces in the Minkowski space
%J Comptes Rendus. Mathématique
%D 2014
%P 1033-1038
%V 352
%N 12
%I Elsevier
%U https://www.numdam.org/articles/10.1016/j.crma.2014.09.003/
%R 10.1016/j.crma.2014.09.003
%G en
%F CRMATH_2014__352_12_1033_0
Kim, Dong-Soo; Kim, Young Ho; Yoon, Dae Won. On standard imbeddings of hyperbolic spaces in the Minkowski space. Comptes Rendus. Mathématique, Tome 352 (2014) no. 12, pp. 1033-1038. doi : 10.1016/j.crma.2014.09.003. https://www.numdam.org/articles/10.1016/j.crma.2014.09.003/

[1] Anciaux, H. Minimal Submanifolds in Pseudo-Riemannian Geometry, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, USA, 2011

[2] Galvez, J.A.; Martinez, A.; Milan, F. Complete constant Gaussian curvature surfaces in the Minkowski space and harmonic diffeomorphisms onto the hyperbolic plane, Tohoku Math. J., Volume 55 (2003) no. 4, pp. 467-476

[3] Hartman, P. On complete hypersurfaces of nonnegative sectional curvatures and constant mth mean curvature, Trans. Amer. Math. Soc., Volume 245 (1978), pp. 363-374

[4] Hano, J.-i.; Nomizu, K. On isometric immersions of the hyperbolic plane into the Lorentz–Minkowski space and the Monge–Ampère equation of a certain type, Math. Ann., Volume 262 (1983) no. 2, pp. 245-253

[5] Kim, D.-S.; Kim, Y.H. Some characterizations of spheres and elliptic paraboloids, Linear Algebra Appl., Volume 437 (2012) no. 1, pp. 113-120

[6] Kim, D.-S.; Kim, Y.H. Some characterizations of spheres and elliptic paraboloids II, Linear Algebra Appl., Volume 438 (2013) no. 3, pp. 1356-1364

[7] Li, A.M. Spacelike hypersurfaces with constant Gauss–Kronecker curvature in the Minkowski space, Arch. Math. (Basel), Volume 64 (1995) no. 6, pp. 534-551

[8] Nomizu, K. Élie Cartan's work on isoparametric families of hypersurfaces, Part 1, Stanford Univ., Stanford, Calif., 1973 (Proc. Symp. Pure Math.), Volume vol. XXVII, Amer. Math. Soc., Providence, RI, USA (1975), pp. 191-200

[9] Nomizu, K. On isoparametric hypersurfaces in the Lorentzian space forms, Jpn. J. Math., Volume 7 (1981) no. 1, pp. 217-226

[10] O'Neill, B. Semi-Riemannian Geometry. With Applications to Relativity, Pure Appl. Math., vol. 103, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983

Cité par Sources :