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)



Ratner theorem and dense geodesic planes in hyperbolic manifolds
I don't see how Ratner's theorem could be used here as I do not see any relation of your setup with **unipotent** flows. On the other hand, a stronger statement actually follows from ergodicity of the **frame flow** (a generalization of the geodesic flows).


In the 3-dimensional case the state space of the frame flow consists of orthonormal triples of tangent vectors $(v\_1,v\_2,v\_3)$, and the dynamics amounts to the parallel transport of the frame $(v\_1,v\_2,v\_3)$ along the geodesic determined by $v\_1$. The frame flow has a natural invariant measure induced by the Riemannian volume on the underlying manifold. For compact hyperbolic 3-manifolds this invariant measure is ergodic, see the paper <http://www.ihes.fr/~gromov/PDF/10[28].pdf> by Brin and Gromov and the references therein.


It implies that (in your notation) for a.e. tangent plane $\pi$ and a.e. tangent line $\ell\subset\pi$ the set $N(\ell)=\{(\exp(p),v\_p):p\in\ell\}\subset N(\pi)$ is dense in the unitary tangent space of $M$. 


