Immersions of the hyperbolic plane Is it possible to isometrically immerse the hyperbolic plane into a compact Riemannian manifold as a totally geodesic submanifold? Any nice examples?
Edit: Although I did not originally say so, I was looking for injective immersions or at least for immersions that do not factor through a covering onto a compact surface. Thank you for your answers and comments, they've been very helpful. 
 A: Here is a general construction.
Take a non-trivial representation of $H=\text{SL}_2(\mathbb{R})$ into a semisimple Lie group $G$, take $K<G$ a maximal compact subgroup and take $\Gamma<G$ an irreducible cocompact lattice. Endow $X=G/K$ with the standard symmetric space structure and consider the image of $H$ in $X$ which is a totally geodesic hyperbolic space. Its image in $\Gamma\backslash X$ will be a totally geodesic immersion of a hyperbolic plane into a compact Riemannian manifold.
Further, if $H$ is not a factor of $G$, up to Baire generically conjugating $\Gamma$ in $G$, we can get that the image of $H$ will be non-compact and if $X$ is of dimension $\geq 5$ (e.g $G=\text{SO}(5,1)$ or $G=\text{SL}_3(\mathbb{R})$) we can get that the immersion is injective (thanks to Ian Agol for correcting an inaccuracy here in a previous version of my answer).
A: Yes, it immerses isometrically into certain solvmanifolds. Take an Anosov map of $T^2$, such as $\left[\begin{array}{cc}2 & 1 \\1 & 1\end{array}\right]$. The mapping torus admits a locally homogeneous metric modeled on the 3-dimensional unimodular solvable Lie group. The matrix has two eigenspaces with eigenvalues $\frac{3\pm\sqrt{5}}{2}$, and the suspensions of lines on the torus parallel to these eigenspaces give immersed manifolds modeled on $\mathbb{H}^2$. If the eigenspace line contains a periodic point of the Anosov map, the mapping torus of it will be an annulus. But otherwise it will be an immersed injective totally geodesic hyperbolic plane, which I think is what you're asking for. 
A: This is an attempt to visualize the answer by Ian Agol. I am not sure it is correct. If it is, it must be a fundamental domain for the group action from that answer, and the surfaces - totally geodesic images of various projective plane embeddings.

A: Definitely the simplest examples are the covering maps from the upperf half space to a compact hyperbolic surface that Bryant mentioned, although these are not injective. A slight variation is to consider the main diagonal embedding $ \imath: \mathbb{H}^2 \to \mathbb{H}^2 \times \mathbb{H}^2$ and then use different projections on each factor. For example, the first can be the orbit projection $\pi : \mathbb{H} \to \mathbb{H}^2/G$ where $G \subset PSL(2, \mathbb{R})$ contains no translations in the real axis and the second could be $ \pi \circ T$ where $T$ is any of such translations. Then $(\pi, \pi \circ T) \circ \imath $ is one to one.
On the other hand, every compact hyperbolic 3-manifold has plenty of totally geodesic immersed $\mathbb{H}^2$ by just projecting totally geodesics $\mathbb{H}^2 \subset \mathbb{H}^3$, as Bryant pointed it is not clear whether  the projection could be made injective. 
This should be a comment but I am not able to comment yet :)
