Let $f : \Bbb R^n \to \Bbb R^n$ be a $C^1$ map such that *it's Derivative at each point $x \in \Bbb R^n$ is an orthogonal matrix* i.e. $Df_x \in O(n,\Bbb R) \text{ , } \forall x \in \Bbb R^n$ . Then prove that $$f(x) = Ox +b \text{ , for some fixed } O \in O(n,\Bbb R) \text{ ,} \forall x \in \Bbb R^n$$

I have no idea about this problem, so couldn't make any attempt.

curveof equal length, but why a line segment? The endpoints are not fixed. $\endgroup$