I am confused by the term "Isometric Embedding". To my knowledge, this refers to a distance preserving map from a space to another (a mapping $f:(E,d_1) \to (F,d_2)$ such that $d_2(f(x_1), f(x_2)) = d_1(x_1, x_2) )$. But I have the following problem :
On one side, I see papers saying that an isometric embedding of a sphere (with its geodesic distance) to an euclidean space cannot exist; e.g., see The Sphere is not Flat by P. L. Robinson.
On the other side, I see the Nash embedding theorem which says that any surface can be embedded into $R^n$ for some $n$.
What didn't I understand ?