I'm working on a model that would require to use vectorial functions of $\mathbb{R}^n \rightarrow \mathbb{R}^n$, such that $\forall x, y \in \mathbb{R}^n$, $\lVert \frac{df(x)}{dx}(y) \lVert\_2 = \lVert y \lVert\_2$, ie with an orthogonal Jacobian.


I can only think of trivial functions (like $f(x) = Ox + c$ for $O$ orthogonal and $c \in \mathbb{R}^n$).


Are there other functions that verify this property? What would it be if we add the constraint $\forall x \in \mathbb{R}^n$, $\lVert f(x) \lVert\_2 = \lVert x \lVert\_2$ ?



functions with orthogonal Jacobian
Such maps are conformal. A theorem of Liouville says that if $n\geq 3$,
the only conformal maps (defined in some region in $R^n$) are Mobius. A Mobius map is a composition of inversions
in spheres. For example $x\mapsto x/|x^2|$ is the inversion in the unit sphere.
Inversions in all spheres generate the Mobius group.


Derivative of a conformal map is a constant times orthogonal.
So if you require it to be orthogonal, you obtain only affine maps. 


Liouville's theorem does not hold in dimension $2$. Conformal maps
in dimension $2$ are complex analytic function whose derivative is 
not equal to zero. Your condition implies that the complex derivative has constant absolute value, so it is constant, and again you obtain an affine map.


Usually they include orientation-reversing maps to the Mobius group, so conformal
maps can be preserving or reversing orientation. 


EDIT. Of course Liouville proved his theorem for sufficiently smooth functions, and the proof is essentially the same as in the answer of David Speyer.
However this theorem holds under much les restrictive assumptions (for some maps differentiable almost everywhere), and this is one of the subjects discussed in the book of Reshetnyak, Stability theorems in geometry and Analysis. 


