Space-like Isoparametric Hypersurfaces in Lorentzian Space Forms

Abstract

The purpose of this paper is to make clear the so-called Nomizu problem, whether it is possible to find examples of space-like isoparametric hypersurfaces in H₁ ⁿ⁺¹ with more than two distinct principal curvatures. It is proved that a space-like isoparametric hypersurface in H₁ ⁿ⁺¹ or S₁ ⁿ⁺¹ can have at most two distinct principal curvatures. The authors present the classification and explicit analytic expressions of such type of isoparametric hypersurfaces.

Keywords

Lorentzian space form / space-like hypersurfaces / isoparametric hypersurface / 53C50 / 53B25

References

[1.] Cartan E. Familles de surfaces isoparamétriques dans les espaces á courbure constante. Ann. Math., 1938, 17: 177-191.

[2.] Nomizu K. On isoparametric hypersurfaces in the Lorentzian space forms. Jpn. J. Math. (N. S.), 1981, 7(1): 217-226.