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Petrunin proves that a metric space $\mathcal{X}$ admits an intrinsicisometry into $\mathbb{E}^n$ if and only if $\mathcal{X}$ is a pro-Euclideanspace of rank at most $n$. He then shows that either case implies that$\mathcal{X}$ has covering dimension $\leq \, n$. In this paper we extend thisresult to include embeddings. Namely, we first prove that any pro-Euclideanspace of rank at most $n$ admits an intrinsic isometric embedding into$\mathbb{E}^{2n+1}$. We then discuss how Petrunin's result implies a partialconverse to this result.

Intrinsic Isometric Embeddings of Pro-Euclidean Spaces