Suppose $\Omega\subset \mathbb R^2$ is a bounded domain with smooth boundary and suppose that $$ F: \Omega \to \Omega,$$ is a diffeomorphism that fixes $\partial \Omega$ (i.e $F|_{\partial \Omega}$ is equal to the identity map) and such that the pull back of the Euclidean metric under $F$, namely $F^\star e$ is again equal to the Euclidean metric $e$. Can we conclude that $F$ is the identity map?