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I know that there is no complete surface embedded in $\mathbb{R}^3$ of constant curvature -$k$ for any $k$. But you can clearly embed the hyperbolic plane (curvature -1) into hyperbolic 3-space (curvature -1). You can also embed Euclidean planes (curvature 0) as horoballs into hyperbolic 3-space (curvature -1)

But can you embed surfaces of curvature -2 into standard, curvature -1 hyperbolic 3-space? In general, when can a manifold of curvature $k$ contain a closed submanifold of curvature $\ell\neq k$? Is the dimension of the submanifold important?

Brian Rushton
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    I perceive curvature as closely related to scale. For the sphere you can obtain any positive curvature value simply by scaling the sphere. So I'd say embedding a curvature $-2$ surface into even a curvature $-1$ plane (not even space) should be trivial by simply changing the metric between models by a constant factor. Am I missing something here? – MvG Mar 20 '13 at 21:29
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    I'd like the curvature to be measured in the target space; by changing the metric, it's changed the problem. – Brian Rushton Mar 20 '13 at 22:39

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This was the topic of Master's thesis of David Brander. I will restate the most relevant parts here. The hyperbolic plane with curvature $-2$ does not embed into the hyperbolic $3$-space of curvature $-1$. It appears to be unknown whether it embeds into $4$-, $5$- or $6$-dimensional space of curvature $-1$. There is an embedding into $\mathbb R^6$, constructed by D. Blanusa in the 1950s, and of course $\mathbb R^6$ embeds into any hyperbolic $7$-space.

More generally, let $Q_c^n$ be the (simply connected) $n$-dimensional space form of curvature $c$. It is easy to embed $Q_c^n$ into $Q_{\tilde c}^{n+1}$ when $c>\tilde c$. The case $c<\tilde c<0$ is much harder. Here is the summary of embedding results for $c<\tilde c<0$:

  • there exists an isometric embedding of $Q_c^n$ into $Q_{\tilde c}^{6n-5}$
  • there is no isometric embedding of $Q_c^n$ into $Q_{\tilde c}^{2n-2}$ (make it $2n-1$ if $n=2$)

And if your manifold is not simply connected, then I don't know how constant curvature can help. You probably need the Nash-Kuiper type embedding. For compact surfaces the embedding dimension was reduced to $5$ by Gromov (Euclidean target). I do not know if the proof extends to hyperbolic target, but in any case you get an embedding into $H^6$.

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