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?

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 $M$? Are there some "ergodic" properties of such construction?

Does Ratner orbit closure theorem apply in this setting? (and what is the precise statemet of Ratner theorem "usable" here? the wiki page http://en.wikipedia.org/wiki/Ratner%27s_theorems is some how vague. The exposition of Tao's blog is nice, http://terrytao.wordpress.com/2007/09/29/ratners-theorems/ but I'm still missing a precise statemet of a suitable version of Ratner theorem)