TITLE: functions with orthogonal Jacobian
QUESTION [5 upvotes]: 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 $\forall x \in \mathbb{R}^n$, $\lVert f(x) \lVert_2 = \lVert x \lVert_2$ ?

REPLY [3 votes]: A geometric proof would be the following: First note that a diffeo with orthogonal Jacobian preserves the length of curves, in particular of straight lines, thus it preserves the Euclidean distance.
From this it follows easily that it maps lines to lines. From the fundamental theorem of affine geometry, it then follows that the diffeo is an affine map, whose linear part is clearly be orthogonal. (Or just use at once the result that a map in $R^n$ that preserves Euclidean distance is the composition of a translation with an orthogonal linear map.)