Isometric embedding

Solution 1:

The usual 2-sphere exists naturally in $\mathbb R^3$, and in general the usual definition of $S^n$ is as a particular subset of $\mathbb R^{n+1}$ with the induced metric. In that case, the identity map is a locally metric-preserving embedding into $\mathbb R^2$, but it doesn't preserve the global distance. To wit, two diametrically opposed points have distance $2$ in $\mathbb R^3$ but distance $\pi$ along geodesics in the sphere itself.

Thus, the natural embedding works as an isometry when we view the two spaces as Riemannian manifolds, but not when we consider them directly as metric spaces. It appears that both kinds of maps can be called "isometric embeddings", but nonetheless they are different concepts.

Solution 2:

The Nash embedding theorem uses a different $d_2$. When you have a submanifold of a Riemann manifold there's an induced Riemann metric, which comes from the inner product on all the tangent spaces of the ambient manifold. Call this metric $d_3$. In the Nash embedding theorem, in your statement above, replace $(F,d_2)$ with $(f(E), d_3)$. That's the kind of isometric embedding this theorem refers to.