  We establish an analogue of Ratner's orbit closure theorem for any connected
closed subgroup generated by unipotent elements in $\operatorname{SO}(d,1)$
acting on the space $\Gamma\backslash \operatorname{SO}(d,1)$, assuming that
the associated hyperbolic manifold $M=\Gamma\backslash \mathbb H^d$ is a convex
cocompact manifold with Fuchsian ends. For $d=3$, this was proved earlier by
McMullen, Mohammadi and Oh. In a higher dimensional case, the possibility of
accumulation on closed orbits of intermediate groups causes very serious
obstacles, and surmounting these via the avoidance theorem (Theorem 7.13) is
the heart of this paper. Our results imply the following: for any $k\ge 1$,
  (1) the closure of any $k$-horosphere in $M$ is a properly immersed
submanifold;
  (2) the closure of any geodesic $(k+1)$-plane in $M$ is a properly immersed
submanifold;
  (3) any infinite sequence of maximal properly immersed geodesic
$(k+1)$-planes intersecting $\operatorname{core} M$ becomes dense in $M$.
