Normal holonomy in Lorentzian space and submanifold geometry

We prove the polarity of the normal holonomy of riemannian submanifolds of lorentzian space.Using this result we prove that, essentially, there is no submanifold of hyperbolic space which admits a parallel normal field ξ ≠ 0 whose shape operator A ξ has constant eigenvalues.We prove the same result for submanifold of euclidean space by regarding them as submanifolds of a horosphere.This is by recovering the zero eigendistribution of A ξ by the normal holonomy of some riemannian submanifold of lorentzian space In particular, this implies that a homogeneous submanifold with parallel mean curvature H must be totally geodesic (the case H = 0 is a consequence of previous results of Di Scala, in the euclidean case, and Di Scala and the first author in the hyperbolic case).We also prove a generalization, using very simple and geometric facts, of the Homogeneous Slice Theorem of Heintze, Thorbergsson and the first author.