A Lorentzian hypersurface will
be called isoparametric if the minimal polynomial of the shape operator is constant.
This allows for complex or non-simple principal curvatures (eigenvalues of the shape
operator). This paper locally classifies isoparametric hypersurfaces in Lorentz
space.
The classification is done by proving Cartan-type identities for the principal
curvatures and showing that the hypersurface can have at most one non-zero real
principal curvature. Standard examples are given in §3 and the main theorems are in
§4.
The hypersurfaces with minimal polynomials (x − a)2 and (x − a)3 are called
generalized umbilical hypersurfaces since they have exactly one principal curvature.
The classification of these hypersurfaces gives insight into principal curvatures
and the effect of the constant principal curvatures on the structure of a
hypersurface.
