This question is a **cross post** from [Math.SE](https://math.stackexchange.com/questions/3091820/every-riemannian-length-structure-on-mathbbrn-is-induced-by-a-continuous-f). Unfortunately the migration of the question is not possible after two months of posting.


I have been reading about *length spaces* in the (great) book Metric Geometry by Y. Burago, D. Burago and S. Ivanov. They define what an *induced length structure* is and they do the following claim without details or references; they just say that it is not easy to prove. It is the Example 2.2.3.



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> **I am looking for a proof of the following claim**: Every riemannian length structure on (the differential manifold) $\mathbb{R}^n$ is induced by a continuous function $f:\mathbb{R}^n\to \mathbb{E}^n$, where $\mathbb{E}^n$ is the euclidean space of dimension $n$. An answer could be (a sketch of) a proof or a reference contaning it.
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It is not clear for me if we can choose the set of admissible paths in $\mathbb{E}^n$ or it is part of the claim that we have to take every continuous path as admissible. At the moment I have no progress in the solution so any remark or comment is also very welcome.


Thanks in advance!


**Relevant definitions:**


A *length structure* on a Hausdorff space $X$ is a pair $(\mathcal{C},\mathcal{L})$, where $\mathcal{C}$ is a set of continuous paths (with closed intervals as domains) in $X$ and $\mathcal{L}$ is a function $\mathcal{C}\to \mathbb{R}\_{\geq0}\cup \{+\infty\}$, satisfying the following axioms.


1) The set $\mathcal{C}$ is closed under restriction (to closed intervals), concatenation and linear reparameterization.


2) The function $\mathcal{L}$ is additive and invariant under linear reparameterizations. It also depends continously on the path in the following sense: If $c:[a,b]\to X$ is in $\mathcal{C}$ then the function $t\in [a,b]\mapsto \mathcal{L}(c|\_{[a,t]})\in \mathbb{R}\_{\geq0}\cup\{+\infty\}$ is continuous.


3) For every $x$ in $X$ there exists an open neightborhood $U\_x$ and a positive real number $R\_x$ such that every path $c$ in $\mathcal{C}$ with $x$ in its image and with image not contained in $U\_x$ verifies that $\mathcal{L}(c)\geq R\_x$.


We call the elements of $\mathcal{C}$ *admissible* paths and the value of $\mathcal{L}$ on such a path is the *length* of the path. A length structure induces a metric on $X$ in which the distance between two points is the infimum of the lengths of admissible paths joining them.


If $f:Y\to X$ is a continuous function between two Hausdorff spaces and $X$ has a length structure $(\mathcal{C},\mathcal{L})$ we define an induced structure $(\mathcal{C}',\mathcal{L}')$ on $Y$ such that a continuous path $c:[a,b]\to Y$ is admissible if and only if its composition with $f$ is admissible and the length of $c$ is equal to the length of such composition. The axiom 3) can fail for this induced structure. If the structure satisfies axiom 3) we call it the *induced length structure*.


A *riemannian length structure* on a differential manifold $M$ is a length structure such that there exists a riemannian metric on $M$ with the same induced metric.


**Suggestions and progress:**


@HKLee has suggested to look in the chapter 6 of [Petrunin and Yashinski](https://www.math.psu.edu/petrunin/papers/mass/euclid2alexandrov.pdf). From there we can extract a proof of the fact that every non-expanding map $g:X\to Y$ between riemannian manifolds of the same dimension can be approximated (maybe uniformly or uniformly on compact sets) by length-preserving maps.



Every riemannian length structure on $\mathbb{R}^n$ is induced by a continuous function $f:\mathbb{R}^n\to \mathbb{E}^n$, to the euclidean space
Well, as far as I can tell, this follows from the Corollary in Section 2.4.11 (p. 216) of Gromov's book *Partial Differential Relations*. I quote it here:



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> Corollary: Let $V$ be an $n$-dimensional stably parallelizable manifold. Then $V$ admits an isometric map $V\rightarrow \mathbb{R}^n$. 
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However, I'm not an expert on the subject, so you should check whether this is really what you need.


