  Given irreducible representations $\Pi$ and $\pi$ of the rank one special
orthogonal groups $G=SO(n+1,1)$ and $G'=SO(n,1)$ with nonsingular integral
infinitesimal character, we state in terms of $\theta$-stable parameter
necessary and sufficient conditions so that \[
\operatorname{Hom}_{G'}(\Pi|_{G'}, \pi )\not = \{0\}. \] In the special case
that both $\Pi$ and $\pi$ are tempered, this implies the Gross--Prasad
conjectures 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 distinguished
representations. If both $\Pi$ and $ \pi$ have the trivial infinitesimal
character $\rho$ then we use a theorem that the periods are nonzero on the
minimal $K$-type to obtain a nontrivial bilinear form on the $({\mathfrak
g},K)$-cohomology of the representations.
