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Consider the hyperbolic $3$-space $\Bbb H^3$ (i.e., the unique, simply-connected, $3$-dimensional complete Riemannian manifold with a constant negative sectional curvature equal to $-1$). The Nash embedding theorem says that $\Bbb H^3$ can be isometrically (i.e., preserving the length of every path) embedded into some $\Bbb R^n$.

What is the minimum $n$ for which $\Bbb H^3$ can be isometrically embedded in $\Bbb R^n$ such that the image of the embedding is bounded?

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    $\begingroup$ H^3 has an isometric embedding into a compact 6-manifold, which in turn has a (continuously differentiable) isometric embedding into R^12 by the Nash-Kuiper theorem. So 12 suffices in the C^1 category. en.m.wikipedia.org/wiki/Nash_embedding_theorems $\endgroup$
    – Ian Agol
    Commented Jan 8, 2023 at 22:10

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