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).
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3$\begingroup$ I suspect you want a subsurface to be totally geodesic; otherwise it is quite useless (you can embed isometrically every surface in every hyperbolic manifold of dimension $\ge 7$ or so). On the positive side, by Kahn-Markovic, there are always almost totally geodesic compact immersed subsurfaces (and their fundamental groups are embedded). But if you want totally geodesic surfaces, there are no good conditions. $\endgroup$– MishaCommented Apr 5, 2019 at 3:37
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$\begingroup$ Thanks Misha! Please can give a reference for the statement about isometric embeddings in dimension $\geq 7$? Also, do you know what happens in dimensions $4,5$ and $6$? $\endgroup$– Nick LCommented Apr 5, 2019 at 3:39
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3$\begingroup$ I think that most of what is known about this kind of question is summarized in the introduction to the following paper: arxiv.org/abs/1802.04619 $\endgroup$– Andy PutmanCommented Apr 5, 2019 at 3:48
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1$\begingroup$ I know very little, beyond the definition, about hyperbolic manifolds, but I do know a lot about isometric embeddings. I agree with Misha and Andy that isometric embeddings without additional assumptions are unlikely to be useful. There are too many degrees of freedom in the embedding, so it's really hard to extract useful geometric information from the embedding. In particular, the key local invariant is the second fundamental form, and in codimensions 2 or more, it is a family of symmetric tensors satisfying the Gauss equations, which is a real algebraic variety and hard to study. $\endgroup$– Deane YangCommented Apr 5, 2019 at 14:39
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1 Answer
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The thing is that every compact Riemannian surface admits a $C^\infty$ isometric embedding in ${\mathbb E}^5$, see Michael Albanese's answer here; 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.
