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.

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, and reflections, which preserve distances between points. They constitute the Euclidean group.

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).

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es