Let $f,g$ and $h$ be in $C^\infty$ on $\mathbb{R}^3$. Is the following statement true?
If $x^2+y^2+z^2=(f(x,y,z))^2+(g(x,y,z))^2+(h(x,y,z))^2$ then $F=(f,g,h)$ is linear
Maybe this could be a consecuence for some conformal map theorem for $\mathbb{R}^n$ with $n\geq 3$? Many thanks!
No. For example, let $$f(x,y,z)=x\sin z-y\cos z$$$$g(x,y,z)=x\cos z+y\sin z$$ $$h(x,y,z)=z$$ Then, $F(1,1,\frac{\pi}{4})=(0,\sqrt2,\frac\pi4)$ and $F(2,2,\pi/2)=(2,2,\frac{\pi}{2})$. Hence $F(2,2,\frac{\pi}{2}) \neq2F(1,1,\frac{\pi}{4})$, which contradicts linearity of $F$.