Suppose we have a closed hyperbolic $3$-manifold $M$. For any $x\in M$ and plane $\pi$ in $T_xM$ we consider $P$ the geodesic plane exp$(\pi)$ originating from $\pi$. For any $p\in \pi$ we consider the unit normal vector $v_p\in T_{exp(p)}M$. Is it true that for gereric $\pi$ the set $N(\pi)=\{(exp(p),v_p): p\in\pi\}$ is dense in the unitary tangent space of $M$? What are the "ergodic" properties of such construction?