Let $(M,g)$ be a compact orientable hyperbolic manifold with dimension at least $4$. Are there any topological conditions on $M$ which guarantee the existence of a hyperbolic surface in $M$? i.e. a $2$-dimensional submanifold $\Sigma$ such that the restriction of $g$ to $\Sigma$ is hyperbolic. (apologies if this is very naïve since this is not my area).



submanifold of a hyperbolic manifold
The thing is that every compact Riemannian surface admits a $C^\infty$ isometric embedding in ${\mathbb E}^5$, see Michael Albanese's answer [here](https://mathoverflow.net/questions/325842/isometric-embedding-of-a-genus-g-surface); the result is a version of the Nash isometric embedding theorem due to Gromov.


Next, ${\mathbb E}^5$ embeds isometrically in ${\mathbb H}^6$ as a horosphere. From this, it follows that for every hyperbolic manifold $M$ of dimension $\ge 6$ and for every compact hyperbolic surface $\Sigma$, there is an isometric null-homotopic immersion $\Sigma\to M$. Such an immersion becomes an embedding after passing to a finite covering space of $M$ (due to the residual finiteness of $\pi\_1(M)$). 


The situation with isometric embeddings in lower dimensions is unclear. 


On the other hand, isometric embeddings like this are pretty useless (as far as I know). More interesting are **totally geodesic** isometric immersions and embeddings. Take a look at the link given by Andy. 


If you are willing to relax the condition to an "$\epsilon$-nearly totally geodesic immersion" (in a suitable sense), then Kahn and Markovic proved that such exist in hyperbolic manifolds of all dimensions with $\epsilon$ as close to $0$ as you wish (at the expense of having high genus). Hope it helps. 


