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A hypersurface of a pseudo-Riemannian manifold is said to be isoparametric if its shape operator has the same characteristic polynomial at all points. Xiao has classified Lorentzian isoparametric hypersurfaces of the anti de Sitter space, $H^n_1$, into four types (see Xi). The classification in two types (I and IV) is up to a global isometry and in the other two types (II , III) up to a local isometry. I am looking for a global description in types II and III under an additional assumption. Precisely
Let $N \subset H^n_1$ be a (connected) extrinsic homogeneous isoparametric hypersurface of type II or III. Is there any global description (up to isometry or diffeomorphism) of $N$?
A sub-question is: Can hypersurfaces of type II and III be extrinsic homogeneous?

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