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.
Suppose that the sphere S n has initially a homogeneous distribution of mass and let G be the Lie group of orientation preserving projective diffeomorphisms of S n . A projective motion of the sphere, that is, a smooth curve in G, is called force free if it is a critical point of the kinetic energy functional. We find explicit examples of force free projective motions of S n and, more generally, examples of subgroups H of G such that a force free motion initially tangent to H remains in H for all time (in contrast with the previously studied case for conformal motions, this property does not hold for H = S O n+1 ). The main tool is a Riemannian metric on G, which turns out to be not complete (in particular not invariant, as happens with non-rigid motions), given by the kinetic energy.
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