Isometry

An isometry between metric spaces preserves every distance. A Riemannian isometry preserves the Riemannian metric and therefore distances, angles, geodesics, and intrinsic curvature.

The isometry group of a metric space consists of its bijective isometries, with composition as the group operation. For a Riemannian metric, these are the diffeomorphisms preserving the metric tensor. Right translations of a right-invariant Riemannian metric supply a faithful subgroup isomorphic to the underlying Lie group.

The fixed-point set of a map $f:M\to M$ is $M^f=\{x:f(x)=x\}$. For an isometry of a Riemannian manifold, each smooth component of the fixed-point set is totally geodesic.

A Euclidean isometry has the form $\mathbf{x}\mapsto \mathbf{a}+R\mathbf{x}$, where $R$ is orthogonal. It preserves inner products of displacement vectors, distances, angles, and arc length.

A glide reflection of the plane composes reflection in a line with a nonzero translation parallel to that line. It reverses orientation and has no fixed point. It cannot be one reflection, because a reflection fixes a line, or a product of two reflections, because such a product preserves orientation. The finite reflection decomposition of a Euclidean isometry therefore shows that it requires exactly three reflections.

Every Euclidean isometry of ${\mathbb{R}}^n$ is a product of at most $n+1$ operations of reflection in a hyperplane. An isometry fixing zero is orthogonal: the polarization identity preserves inner products, and its values on an orthonormal basis determine its linear action. An orthogonal transformation is a product of at most $n$ linear reflections, by reflecting the image of the first basis vector back to that vector and inducting on its orthogonal complement. For an arbitrary isometry, first reflect its image of zero back to zero, if necessary, and apply the orthogonal result. Affine reflection hyperplanes, rather than only hyperplanes through zero, are essential here.

An isometric embedding $f:X\to Y$ between metric spaces is an injective function satisfying $d_Y(f(x),f(x^{\prime }))=d_X(x,x^{\prime })$ for all $x,x^{\prime }\in X$.

An isometric extension of a connected metric manifold is a proper embedding onto an open subset of a larger connected manifold of the same dimension, preserving the metric tensor. Smooth complete connected Riemannian manifolds admit no such proper extension; Lorentzian geodesic completeness has different subtleties.