Equidistribution of expanding translates of smooth curves in homogeneous spaces under the action of a product of SO(n,1)'s

We study the limiting distributions of expanding translates of a compact segment of a smooth curve under a diagonal subgroup of $G=\prod_{i=1}^k\mathrm{SO}(n_i,1)$, where $G$ acts on a finite volume homogeneous space $L/Γ$ as a subgroup. We show that the expanding translates of the curve become equi