We are talking about surfaces in $\mathbb{R}^3$ here. I know that not every isometry from a surface to another is a congruence. But what about isometries from a surface to itself? Can someone give an example that is not a congruence?
Isometry in differential geometry is different from that in geometry. Here it means a diffeomorphism that preserves the first fundamental form. A congruence is a rigid motion.