Does a map that preserves distance imply isometry?

Question

It feels intuitive for this to be true. Let us say I have a chart from $\mathbb{R^2}$ to some section of a manifold in $\mathbb{R^3}$. If the distance in $\mathbb{R^2}$ between two points is equal to the shortest distance (locally, the geodesic) on the manifold in $\mathbb{R^3}$, does this imply the map is an isometry?