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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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    "agrees with a Euclidean reflection on the curved sides of the cone": I meant cylinder, not cone.2012-05-06

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