Isometry

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

A transformation preserving a spacetime metric tensor. For Minkowski spacetime, the affine transformations with this property form the Poincaré group. A spacetime isometry is a symmetry of particular dynamical equations only if their other fixed structures and couplings are preserved as well.

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.

A finite group of orientation-preserving isometries of the Euclidean plane or hyperbolic plane is a cyclic group. Take one orbit and its unique minimum-radius enclosing disc. Every group element preserves the orbit and therefore the disc's centre. The orientation-preserving stabilizer of a point acts as planar rotations, so the group embeds in the circle group. In a nontrivial finite subgroup, the least positive rotation angle generates all angles by division with remainder, proving cyclicity. Orientation reversal changes the conclusion: reflection groups need not be cyclic.

For a connected three-dimensional Riemannian manifold, the Myers-Steenrod theorem bounds the dimension of its isometry group by six. Dimension five cannot occur: isotropy embeds in $O(3)$, whose Lie subalgebras have dimensions zero, one or three. A five-dimensional action would require three-dimensional isotropy and a two-dimensional orbit; full rotational isotropy has no invariant tangent two-plane. The maximal dimension six forces transitivity and isotropy, hence constant sectional curvature. Each of the six listed dimensions is attained by a suitable metric.

For a complete simply connected Riemannian manifold $X$ and a free properly discontinuous deck group $\Gamma$, each isometry of $X/\Gamma$ lifts to an isometry of $X$ normalizing $\Gamma$. Conversely such a normalizing isometry descends; two lifts differ by a deck transformation. This proves the formula, including orientation-reversing isometries if the full normalizer is used.

The isometry group of a smooth connected Riemannian manifold is a finite-dimensional Lie group, and its action is smooth. An isometry is determined by its value and derivative at one point, since it preserves the exponential map and continuation along geodesics. The stabilizer embeds in $O(n)$, giving dimension at most $n(n+1)/2$. For a compact manifold the isometry group is compact.

A finite group of Euclidean isometries has a common fixed point: average any orbit. Each isometry is an affine map and therefore preserves this finite affine average; applying a group element merely permutes the orbit terms. To see that an isometry is affine, subtract its value at zero and use the polarization identity to show that the resulting map preserves inner products. Images of a standard orthonormal basis then determine a linear orthogonal map.

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.

The vertex set of a triangle with three distinct side lengths has trivial setwise stabilizer subgroup in the plane isometry group. Each vertex is characterized by the unordered pair of lengths of its incident edges, so a setwise-preserving Euclidean isometry fixes every vertex. An isometry fixing three noncollinear plane points is the identity: squared Euclidean distances to them determine every other point, since subtracting two of the Euclidean distance equations gives two independent linear equations. For example, $(0,0),(1,0),(0,2)$ have side lengths $1,2,\sqrt{5}$.

In Euclidean geometry, a rotation about the origin is an orientation-preserving orthogonal linear map, represented by an element of the special orthogonal group. About another center $a$, the map has form $x\mapsto a+R(x-a)$. In three dimensions a nonidentity rotation fixes an axis.

The fixed straight line of a nonidentity three-dimensional rotation. For a rotation matrix $R$, its direction vector is an eigenvector with eigenvalue $1$. A unit vector along it also specifies the positive sense of the angle through the right-hand convention.

A rotation through an angle of $\pi$. In three dimensions, a half-turn about a straight line with unit vector $\mathbf{n}$ fixes its parallel component and negates its orthogonal complement, giving the rotation matrix $2\mathbf{n}{\mathbf{n}}^T-I$ by orthogonal projection.

The translation subgroup of a group of Euclidean isometries consists of its translations. It is a normal subgroup, because conjugation by $x\mapsto Ax+b$ takes translation by $v$ to translation by $Av$. Its vectors form an additive subgroup of Euclidean space; for a plane crystallographic group they form a rank-two translation lattice.

A translation lattice is the additive group generated by linearly independent vectors $v_1,\dots ,v_r$ in a Euclidean space. It defines a discrete subgroup of translations. A full-rank translation lattice has a compact parallelepiped as a fundamental region and therefore a cocompact group action. This geometric usage is distinct from an order-theoretic lattice.

A discrete subgroup of Euclidean isometries is a frieze group when its translation subgroup is infinite cyclic and has finite index. Translations along a line alone give $\mathbb{Z}$; adding perpendicular mirrors gives the infinite dihedral group, while adding a mirror along the translation direction gives $\mathbb{Z}\times C_2$. These groups describe patterns periodic in one planar direction.

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.

A product of $m$ reflections has $\mathrm{rank}(I-Q)\le m$, by the rank inequality applied successively to $I-AB=(I-A)+A(I-B)$. Conversely an orthogonal transformation fixes the orthogonal complement of its moved space; successive Householder reflections fix one new basis direction at each step in that moved space. At most its dimension $\mathrm{rank}(I-Q)$ are required. Thus central inversion in three dimensions needs exactly three reflections.

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.