Lorentzian isoparametric hypersurfaces in H₁ⁿ⁺¹

Liang Xiao

Vol. 189 (1999), No. 2, 377–397

DOI: 10.2140/pjm.1999.189.377

Abstract

In 1981, Nomizu introduced isoparametric hypersurfaces in Lorentzian space forms and studied the Cartan identities. Later Hahn, 1984, generalized Nomizu’s work to the pseudo-Riemannian space forms and presented many examples. In general, the shape operator of a hypersurface in a pseudo-Riemannian space form may be not diagonalizable. This makes the isoparametric theory in pseudo-Riemannian space form different from that in Riemannian space forms. In 1985, Megid classified Lorentzian isoparametric hypersurfaces in R₁ⁿ⁺¹. He showed that there are three types of Lorentzian isoparametric hypersurfaces in R₁ⁿ⁺¹. Type I are exactly cylinders and umblic hypersurfaces while the other two types of hypersurfaces have properties close to cylinders and umblic hypersurfaces. Megid called them generalized cylinders and umblic hypersurfaces. In this paper, the local classification of Lorentzian isoparametric hypersurfaces in H₁ⁿ⁺¹ is obtained and the properties of them are discussed.

Milestones

Received: 23 November 1997

Revised: 15 May 1998

Published: 1 June 1999