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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)

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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 $M$.

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    $\begingroup$ the statement of ratner theorem in the Tao blog applies, as $PSL(2,R)$ is generated by unipotents. and a plane in $H^3$ is the orbit of a $PSL(2,R)$ action $\endgroup$
    – user126154
    Commented Jul 17, 2014 at 16:13
  • $\begingroup$ one does not need Ratner's here as you don't need MC (or the topological orbit closure theorem), as you are speaking about a "generic" orbit, which is equidistributed by simple ergodicity (say Howe-Moore). Notice that the unitary space is actually a Torus bundle over the usual homogeneous space (think of ASL2 instead of SL2) and there are slight modifications (by the duality principle) that will give you the equidistribution in that space (see the Elkies-McMullen paper or the new work by Strombergsson, or Eskin's Pisa notes). $\endgroup$
    – Asaf
    Commented Jul 17, 2014 at 20:47

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