We prove here, by geometric, or rather dynamical, methods, the following theorem. Let G be a non-compact connected Lie subgroup of the isometry group Isom (H-n) of the real hyperbolic space H-n, which does not fix any point at infinity, i.e. on partial derivativeH(n) similar or equal to Sn-1. Then G preserves a certain hyperbolic subspace H-d subset of W and 'contains' all the identity components Isom(0)(H-d) of its isometry group. We provide an 'algebra-free' proof and present the dynamical tools used, so that the exposition is 'self-contained'.
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