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. 



Immersions of the hyperbolic plane
Yes, it immerses isometrically into certain [solvmanifolds.](https://en.wikipedia.org/wiki/Solvmanifold) 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](https://books.google.com/books?id=5irlDQAAQBAJ&lpg=PP1&dq=thurston%20geometry%20and%20topology&pg=PA188#v=onepage&q&f=true). 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. 


