Is there a function $f:\mathbb{R}^n \to \mathbb{R}^n$ with derivative $Df(x_0) \in SO_n(\mathbb{R})$ for every point $x_0 \in \mathbb{R}^n$, which is not affine linear? I am interested in two cases: (i) if $f$ is $C^1$ (ii) if $f$ is smooth
For $n=1$ the answer is no trivially, since $SO_1(\mathbb{R})=\{1\}$. For $n=2$ we can translate the question into one in complex analysis: is there an analytic (entire) function $f$ with $|f'|=1$ everywhere which is not linear, to which the answer is no again. However for $n\geq 3$ I am not sure how to proceed...
Another approach comes to mind: such a function $f$ must satisfy a system of PDEs which basically says $Df(x_0) \in SO_n(\mathbb{R})$. For example for $n=2$ the system would be $$(\frac{\partial f_1}{\partial x})^2+(\frac{\partial f_1}{\partial y})^2=1$$ $$(\frac{\partial f_2}{\partial x})^2+(\frac{\partial f_2}{\partial y})^2=1$$ $$\frac{\partial f_1}{\partial x}\frac{\partial f_2}{\partial x}+\frac{\partial f_1}{\partial y}\frac{\partial f_2}{\partial y}=0$$ $$\frac{\partial f_1}{\partial x}\frac{\partial f_2}{\partial y}-\frac{\partial f_1}{\partial y}\frac{\partial f_2}{\partial x}=1$$ Sadly I know little about PDEs and have no idea how to determine whether there exists non-linear solutions to such systems...
I would be grateful if someone can help shed light on this question, not necessarily following what I mentioned above. Thanks!