All the references I have seen so far list the Nash $C^1$-embedding theorem as an example where the $h$-principle holds. The $h$-principle for a differential relation holds by definition, when the inclusion Solutions -> "Relaxed Solutions" induces a surjective map on the level of fundamental groups.
Nash proves that one can find in every $\varepsilon$-neighborhood of a short embedding a $C^1$-isometric embedding. Is it then obvious that one can find also a homotopy between this actual isometric embedding starting from the short embedding?
In my opinion the statement with the $\varepsilon$-closeness and the homotopy are not equivalent or am I missing something?