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

Question

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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.

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.

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. 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.

Answer 1

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:

Corollary: Let $V$ be an $n$-dimensional stably parallelizable manifold. Then $V$ admits an isometric map $V\rightarrow \mathbb{R}^n$.

However, I'm not an expert on the subject, so you should check whether this is really what you need.

Answer 2

I'm not sure I've understood your question but if you meant how one can view a Riemannian manifold as a length space, I've seen that a little bit different in Metric spaces of non-positive curvature textbook ( Bridson and Haefliger ). Specifically,3.18 proposition which claimes let $X$ be a connected Riemannian manifold, given $ x,y \in X $, let $ d(x,y) $ be the infimum of the Riemannian length of piecewise continuously differentiable paths $ c \colon [0,1] \to X $ such that $c(0)=x , c(1)=y $.Then $d$ is a metric on $ X$, The topology defined on $X$ is the same as the given topology on manifold $X$, and also $(X,d)$ is a length space. It is said as an example, in a special case that Riemannian manifold is $ \mathbb{R}^n$, with Riemannian metric $ds^2$, the associated length space is $ \mathbb{E}^n$.