Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Isometric Embeddings of Polyhedra into Euclidean Space   In this paper we consider piecewise linear (pl) isometric embeddings of
Euclidean polyhedra into Euclidean space. A Euclidean polyhedron is just a
metric space $\mathcal{P}$ which admits a triangulation $\mathcal{T}$ such that
each $n$-dimensional simplex of $\mathcal{T}$ is affinely isometric to a
simplex in $\mathbb{E}^n$. We prove that any 1-Lipschitz map from an
$n$-dimensional Euclidean polyhedron $\mathcal{P}$ into $\mathbb{E}^{3n}$ is
$\epsilon$-close to a pl isometric embedding for any $\epsilon > 0$. If we
remove the condition that the map be pl then any 1-Lipschitz map into
$\mathbb{E}^{2n + 1}$ can be approximated by a (continuous) isometric
embedding. These results are extended to isometric embedding theorems of
spherical and hyperbolic polyhedra into Euclidean space by the use of the
Nash-Kuiper $C^1$ isometric embedding theorem. Finally, we discuss how these
results extend to various other types of polyhedra.

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The categories of this paper are: hep-th
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The categories of this paper are: physics
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The categories of this paper are: chem-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: solv-int
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-ph
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The categories of this paper are: eess
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The categories of this paper are: nlin
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The categories of this paper are: acc-phys
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The categories of this paper are: gr-qc
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The categories of this paper are: math-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: q-fin
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The categories of this paper are: cs
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The categories of this paper are: cond-mat