Let $m < n \in \mathbb{N}$. Let us equip $\mathbb{R}^m, \mathbb{R}^n $ with their canonical Euclidean (Riemannian) metrics. How can we characterize the isometric embeddings of $\mathbb{R}^m$ into $\mathbb{R}^n $? To be more precise, I'm looking for sufficiently regular, injective, distance preserving transformations from $\Phi: \mathbb{R}^m \to \mathbb{R}^n$, so that: $d_m(x, y)= d_n(\Phi(x), \Phi(y))$, where $d_m, d_n$ represents the distances in $m, n$ dimensional Euclidean spaces respectively, so e.g. $d_m(x,y):=||x-y||_{\mathbb{R}^m}$.
I think the answer is:
$$x \mapsto A(Bx, O(Bx))$$ where $A$ is an Euclidean isometry of $\mathbb{R}^n$ (i.e. a rigid motion), $B$ is an Euclidean isometry of $\mathbb{R}^m$ (i.e. a rigid motion), and $O:\mathbb{R}^m \to \mathbb{R}^n$ is "zero padding $n-m$" times, namely: $O(x)= (x, 0,\dots 0)$.
If the above correct/incorrect, how do I go about proving it or characterizing the Euclidean embeddings?