Normal Holonomy of Complex Submanifolds 231
Exercise 7.6.6 Extend Lemma 4.5.5 and Theorem 4.5.4 for:
(i) Irreducible Riemannian submanifolds of Lorentzian space, which are con-
tained in a hyperbolic space and not contained in a horosphere.
(ii) Riemannian submanifolds of Lorentzian space, which are contained in a horo-
sphere and are irreducible when regarded as submanifolds of the Euclidean
space.
[Hint: In order to use Exercise 7.6.3 we may assume, by adding a suitable multiple
of the position vector, that the parallel isoparametric n ormal field is timelike.]
Exercise 7.6.7 Prove Theorem 4.5.9 and Theorem 4.5.10. [Hint: Use that there are
no full irreducible proper isoparametric submanifolds of hyperbolic space; see The-
orem 4.2.21 (and the same is true for the Euclidean space, unless the submanifold is
contained in a sphere).]
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