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Given irreducible representations $\Pi$ and $\pi$ of the rank one specialorthogonal groups $G=SO(n+1,1)$ and $G'=SO(n,1)$ with nonsingular integralinfinitesimal character, we state in terms of $\theta$-stable parameternecessary and sufficient conditions so that \[\operatorname{Hom}_{G'}(\Pi|_{G'}, \pi )\not = \{0\}. \] In the special casethat both $\Pi$ and $\pi$ are tempered, this implies the Gross--Prasadconjectures for tempered representations  of $SO(n+1,1) \times SO(n,1)$ which are nontrivial on the center.  We apply these results to construct nonzero periods and distinguishedrepresentations. If both $\Pi$ and $ \pi$ have the trivial infinitesimalcharacter $\rho$ then we use a theorem that the periods are nonzero on theminimal $K$-type to obtain a nontrivial bilinear form on the $({\mathfrakg},K)$-cohomology of the representations.

Distinguished representations of SO(n+1,1) x SO(n,1), periods and branching laws