Null Screen Isoparametric Hypersurfaces in Lorentzian Space Forms

  • Matias NavarroEmail author
  • Oscar Palmas
  • Didier A. Solis


In this paper, we develop the notion of screen isoparametric hypersurface for null hypersurfaces of Robertson–Walker spacetimes. Using this formalism we derive Cartan identities for the screen principal curvatures of null screen isoparametric hypersurfaces in Lorentzian space forms and provide a local characterization of such hypersurfaces.


Null hypersurfaces isoparametric hypersurfaces 

Mathematics Subject Classification

Primary 53B30 Secondary 53C50 



The first author is grateful with CONACYT and Facultad de Ciencias, UNAM for the financial support and warm hospitality during the sabbatical year in which this work was developed. The authors thank the anonymous referee for his/her comments.


  1. 1.
    Alías, L.J., Romero, A., Sánchez, M.: Spacelike hypersurfaces of constant mean curvature and Calabi–Bernstein type problems. Tohoku Math. J. (2) 49(3), 337–345 (1997)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Atindogbe, C., Harouna, M.M., Tossa, J.: Lightlike hypersurfaces in Lorentzian manifolds with constant screen principal curvatures. Afr. Diaspora J. Math 16(2), 31–45 (2014)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Besse, A.L.: Einstein Manifolds. Classics in Mathematics. Springer, Berlin (2008) (Reprint of the 1987 edition) Google Scholar
  4. 4.
    Cartan, É.: Familles de surfaces isoparamétriques dans les espaces à courbure constante. Ann. Mat. Pura Appl. 17(1), 177–191 (1938)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cartan, É.: Sur des familles remarquables d’hypersurfaces isoparamétriques dans les espaces sphériques. Math. Z. 45, 335–367 (1939)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cartan, É.: Sur des familles d’hypersurfaces isoparamétriques des espaces sphériques à 5 et à 9 dimensions. Univ. Nac. Tucumán. Revista A. 1, 5–22 (1940)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Cecil, T.E., Chi, Q.S., Jensen, G.R.: Isoparametric hypersurfaces with four principal curvatures. Ann. Math. 166, 1–76 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Cecil, T.E., Ryan, P.J.: Geometry of Hypersurfaces. Springer Monographs in Mathematics. Springer, New York (2015)Google Scholar
  9. 9.
    Chi, Q.S.: Isoparametric hypersurfaces with four principal curvatures, iii. J. Differ. Geom. 94, 469–504 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    d’Inverno, R.: Introducing Einstein’s Relativity. The Clarendon Press, Oxford University Press, New York (1992)zbMATHGoogle Scholar
  11. 11.
    Dong, J., Liu X.: Totally umbilical lightlike hypersurfaces in Robertson-Walker spacetimes. ISRN Geom., pages Art. ID 974695, 10 (2014)Google Scholar
  12. 12.
    Duggal, K.L., Bejancu, A.: Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications. Mathematics and its Applications, vol. 364. Kluwer Academic Publishers Group, Dordrecht (1996)Google Scholar
  13. 13.
    Duggal, K.L., Sahin, B.: Differential Geometry of Lightlike Submanifolds. Frontiers in Mathematics. Birkhäuser, Basel (2010)Google Scholar
  14. 14.
    Gutiérrez, M., Olea, B.: Totally umbilic null hypersurfaces in generalized Robertson–Walker spaces. Differential Geom. Appl. 42, 15–30 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Hahn, J.: Isoparametric hypersurfaces in the pseudo-Riemannian space forms. Math. Z. 187(2), 195–208 (1984)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space-Time. Cambridge University Press, London-New York. Cambridge Monographs on Mathematical Physics, No. 1 (1973)Google Scholar
  17. 17.
    Kang, T.H.: On lightlike hypersurfaces of a GRW space-time. Bull. Korean Math. Soc. 49(4), 863–874 (2012)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Laura, E.: Sopra la propagazione di onde in un mezzo indefinito. Scritti Matematici Offerti ad Enrico D’Ovidio, pp. 253–278 (1918)Google Scholar
  19. 19.
    Levi-Civita, T.: Famiglie di superficie isoparametriche nell’ordinario spazio euclideo. Rend. Acc. Naz. Lincei. 26, 335–362 (1937)zbMATHGoogle Scholar
  20. 20.
    Li, C., Wang, J.: The classification of isoparametric surfaces in \({{\mathbb{S}}}^3_1\). Kobe J. Math. 22(1–2), 1–12 (2005)MathSciNetGoogle Scholar
  21. 21.
    Li, Z., Xie, X.: Spacelike isoparametric hypersurfaces in lorentzian space form. Front. Math. China 1, 130–137 (2006)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Magid, M.A.: Lorentzian isoparametric hypersurfaces. Pacific J. Math. 118(1), 165–197 (1985)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Miyaoka, R.: Isoparametric hypersurfaces with \((g, m) = (6, 2)\). Ann. Math. 177, 53–110 (2013)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Montiel, S.: Real hypersurfaces of a complex hyperbolic space. J. Math. Soc. Jpn. 37, 515–535 (1985)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Münzner, H.F.M.: Isoparametrische hyperfächen in sphären. I. Math. Ann. 251, 57–71 (1980)CrossRefGoogle Scholar
  26. 26.
    Münzner, H.F.M.: Isoparametrische hyperfächen in sphären. II. Math. Ann. 256, 215–232 (1981)CrossRefGoogle Scholar
  27. 27.
    Navarro, M., Palmas, O., Solis, D.A.: Null screen quasi-conformal hypersurfaces in semi-Riemannian manifolds and applications (in preparation) Google Scholar
  28. 28.
    Navarro, M., Palmas, O., Solis, D.A.: Null hypersurfaces in generalized Robertson–Walker spacetimes. J. Geom. Phys. 106, 256–267 (2016)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Nomizu, K.: On isoparametric hypersurfaces in the Lorentzian space forms. Japan. J. Math. (N.S.) 7(1), 217–226 (1981)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Barrett, O.: Semi-Riemannian Geometry, with Applications to Relativity, volume 103 of Pure and Applied Mathematics. Academic Press, Inc., (1983)Google Scholar
  31. 31.
    San Martín-López, V.: Spacelike isoparametric hypersurfaces. Differ. Geom. Appl., 203–207 (2017) (to appear) Google Scholar
  32. 32.
    Segre, B.: Famiglie di ipersuperficie isoparametriche negli spazi euclidei ad un qualunque numero di dimensioni. Rend. Acc. Naz. Lincei. 27, 203–207 (1938)zbMATHGoogle Scholar
  33. 33.
    Somigliana, C.: Sulle relazione fra il principio di huygens e l’ottica geometrica. Atti Matematici Acc. Sc. Torino, 54, 974–979 (1918–1919)Google Scholar
  34. 34.
    Takagi, R.: Real hypersurfaces in a complex projective space with constant principal curvatures i. J. Math. Soc. Jpn. 27, 43–53 (1975)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Wald, R.M.: General Relativity. University of Chicago Press, Chicago, IL (1984)CrossRefGoogle Scholar
  36. 36.
    Xiao, L.: Lorentzian isoparametric hypersurfaces in \({{\mathbb{H}}}^{n+1}_1\). Pacific J. Math. 189(2), 377–397 (1999)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Yau, S.T.: Open problems in geometry. In S.T. Yau, editor, Chern—A Great Geometer of the Twentieth Century, pp. 275–319. International Press (1992)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Facultad de MatemáticasUniversidad Autónoma de Yucatán-UADYMéridaMexico
  2. 2.Departamento de Matemáticas, Facultad de CienciasUniversidad Nacional Autónoma de México-UNAMMexico CityMexico

Personalised recommendations