Totally geodesic hypersurface in compact hyperbolic manifold In [Zeghib: Laminations et hypersurfaces géodésiques des variétés hyperboliques, Annales scientifiques de l'ENS, 1991] it is shown, that in a compact manifold of negative curvature, there exists only a finite number of totally geodesic hypersurfaces (codim=1) without self-intersections.
Now my question: Is there an example of such a compact manifold M (with dim(M) > 2)  , that allows even one totally geodesique hypersurface? 
Or a result that proofs the existence?
In manifolds of variable curvature it is not clear, that there exists even one totally geodesic hypersurface.
In the case of constant curvature it is clear in the universal covering (hyperbolic space), bur for a compact manifold?
 A: Examples in dimension $3$ follow from hyperbolization theorems for 3-manifolds combined with Mostow rigidity. Suppose that you have a compact 3-manifold $M$ with connected boundary $\partial M$ of genus $\ge 2$, such that $M$ is irreducible, atoroidal, and acylindrical and $\partial M$ is incompressible --- in brief this says $M$ has no "badly embedded" spheres, discs, annuli, or tori. 
Now produce a closed 3-manifold $DM$, the "double" of $M$, by taking the quotient of two copies of $M$ identifying their boundaries by the "identity map". It follows that $DM$ is irreducible, atoroidal, and has infinite fundamental group. 
The closed 3-manifold $DM$ has a hyperbolic structure; you can use Thurston's original hyperbolization theorem for this, because $\partial M$ becomes an incompressible surface in $DM$. 
By the Mostow rigidity theorem, the "reflection" map from $DM$ to itself, interchanging the two copies of $M$ and being the identity on $\partial M$ itself, is an isometry. Therefore, $\partial M$ is totally geodesic in $DM$.
