Isometry

In the sense of @needham2021visual, an isometry of a surface $S$ into other $\tilde{S}$ is a map that preserves distances and angles, in the sense of abstract manifolds. If it preserves orientation it is called a direct isometry.

In general, given two pseudo-Riemannian manifolds $(M,g)$ and $(N,h)$ and a diffeomorphism $f:M\to N$, $f$ is called isometry (or isometric isomorphism) if $g=f^{\ast }(h)$.
If $f$ is a local diffeomorphism then it is called local isometry.

Related: symmetry of a pseudo-Riemannian manifold.

The Euclidean plane

In the special case of Euclidean space, which is a very simple example of a Riemannian manifold, an isometry is often called a "rigid motion". These include translations, rotations, reflections and glide reflections, which preserve distances between points. They constitute the Euclidean group.

Steps to "see" the isometry group of the Euclidean plane

1. The geodesics of this Riemannian manifold can be shown to be straight lines.
2. Given two points, there is only one straight line joining them.
3. The geodesic distance can be explicitly proven to be $$d(p,q)=\sqrt{(q_1-p_1)^2+(q_2-p_2)^2}$$ which is the the Euclidean norm formula.
4. Isometries preserve geodesics, so they preserve this distance. We are now in the context of an Euclidean affine space.
5. In the context of an affine space, it is well-known that the distance-preserving maps are only translations, rotations, and reflections. Is basic linear algebra.
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Surfaces in ${\mathbb{R}}^3$

Having said that, given two surfaces in ${\mathbb{R}}^3$ we have to distinguish between an isometry sending the first one to the second one (their first fundamental forms agree) from a congruence: an isometry of ${\mathbb{R}}^3$ sending the first surface to the second one. Gauss's Theorema Egregium says that isometric surfaces have the same Gaussian curvature, but the converse is not true: there are examples of surfaces with the same Gaussian curvature, but which are not isometric.

A counterexample for that is the exponential horn ($X_1(u,v)=(u\cos v,u\sin v,\log u)$) and the cylinder ($X_2(u,v)=(u\cos v,u\sin v,v)$), which have same Gaussian curvature at corresponding points, but are actually not isometric (calculate the first fundamental form and see that they are essentially different).