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@AntonPetrunin Hi, I took a look to your notes and I think they are a very good introduction. Since they were already mentioned in another comment, I have added them to my answer to make them easier to find.
I had a look to that book and it seems that you are right. Although it will take me some time to understand the details. Feel free to post again your answer with your main account. Thanks!
What you say is completely true, but I am sorry to tell you that you misunderstood the question. Anyway thank you for your time! The question is about how to build a given Riemannian metric on $ \mathbb {R} ^ n $ using an appropriately chosen continuous function $f:\mathbb{R}^n\to\mathbb{E}^n$ (and possibly an appropriately chosen family of admissible paths in $\mathbb{R}^n$) in the way detailed in the question. For example, we could ask how to get the hyperbolic $n$-space $\mathbb{H}^n$ with such a setting.
