I am differential geometry beginner. I have one question.
Following statement is well known. If $F$ : $\Bbb R^m\rightarrow\Bbb R^n$ is a map & diffeomorphism. And if $F_*$(tangent map of $F$) preserves inner product, then $F$ is an isometry.
But In $m=n=3$, $F$ is still Isometry even if we subtract the condition of '$F$ is a diffeomorphism'
I tried several ways to prove this, but it was useless. I really want to know this elementary proof. Thanks.