Wikipedia states the hypothesis of the Nash-Kuiper theorem as follows:
Let $(M,g)$ be an $m$-dimensional Riemannian manifold and $f: M \to \mathbb{R}^n$ a short $\mathcal{C}^\infty$-embedding (or immersion) into Euclidean space $\mathbb{R}^n$, where $n ≥ m+1$.
We usually either talk about (a) "embeddings" between differentiable manifolds or (b) "isometric embeddings" between Riemannian manifolds. So if someone talks about just an "embedding" between Riemannian manifolds, is it implied that this embedding is isometric, or does the embedding only apply to the underlying differentiable manifold structure and not to the metric structure?
I assume that $f$ isn't required to be isometric, because if it is then the Nash-Kuiper theorem seems totally trivial, because you can just let $f_\epsilon$ be $f$ itself.