Let $\Bbb F :\Bbb R^n\rightarrow\Bbb R^3$ is a map. And $\Bbb F_*$ be the tangent map of $\Bbb F$
If $\Bbb F$ is diffeomorphism and $\Bbb F_*$ preserves inner product, then we know that $\Bbb F$ is isometry.
Question is: If excepting one condition that $\Bbb F$ is diffeomorphism, then $\Bbb F$ can be not an isometry?
I can find example for $\Bbb F :\Bbb R^2\rightarrow\Bbb R^3$ is not isometry, but I can't find example for when domain is $\Bbb R^3$
I'm waiting for your help. Thanks