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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.

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what do you mean by congruence.. WIKIPEDIA SAYS THAT:In geometry, two figures are congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object. More formally, two sets of points are called congruent if, and only if, one can be transformed into the other by an isometry, i.e., a combination of translations, rotations and reflections. – zapkm May 5 '12 at 18:30
Take a cylinder capped at either end by a disk. In the middle of the disk, there is a cone. At one end the cone sticks in, and at one end the cone sticks out. Now you can have an isometric involution of this surface that agrees with a Euclidean reflection on the curved sides of the cone, but the Euclidean reflection does not give an isometry of the surface. No Euclidean isometry can. – yasmar May 5 '12 at 19:23
"agrees with a Euclidean reflection on the curved sides of the cone": I meant cylinder, not cone. – yasmar May 6 '12 at 10:09

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