Differential Geometry: Lecture 22: congruence of surfaces

Two curves are congruent if they have same unit-speed and matching curvature and torsion. In the same sense, two surfaces are congruent if they are isometric and have matching shape operators. I leave the proofs to O'neill, but I hope you can appreciate the distinction between extrinsic and intrinsic a bit more from this discussion. This is based on Section 6.9 of O'neill.