Isometry Groups and Homogeneity

Each of the three constant-curvature geometries is completely captured by its group of isometries — the distance-preserving transformations that formalize Euclid's "superposition" and define what "congruent" means. This page makes precise the group column of the space-form table: the Euclidean group $E(n)$, the spherical group $O(n+1)$, and the hyperbolic group $O(n,1)$. It presents each geometry as a homogeneous space $G/H$, and it draws the bridge that matters most for the rest of the book — the hyperbolic isometry group $O(n,1)$ is the Lorentz group of special relativity. It reuses the Lie-group and semidirect-product machinery of group-theory/README.md.

References: Ratcliffe, Foundations of Hyperbolic Manifolds; Thurston, Three-Dimensional Geometry and Topology, ch. 2; Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces; Berger, Geometry I.

1. Isometries and Congruence

An isometry of a Riemannian space $M$ is a bijection $f:M\to M$ preserving distance, $d(f(p),f(q))=d(p,q)$. The isometries form a group $\mathrm{Isom}(M)$ under composition, and — since a geodesic is determined by metric data — isometries send geodesics to geodesics and preserve angles. Two figures are congruent exactly when some isometry maps one onto the other, so $\mathrm{Isom}(M)$ is the group of congruences: the group-theoretic distillation of the whole synthetic theory. As established for the plane in euclidean-results.md, and true in all three geometries (the Cartan–Dieudonné theorem),

Reflections generate. Every isometry of a constant-curvature space is a composition of finitely many reflections in geodesic hyperplanes (at most $n+1$ in dimension $n$). The reflection is the atom of congruence.

2. The Three Isometry Groups

Each space form has the maximal-symmetry isometry group of dimension $\left(\genfrac{}{}{0ex}{}{n+1}{2}\right)$ — enough to move any point to any other (homogeneity) and rotate any frame to any other at a point (isotropy).

2.1 Euclidean: $E(n)={\mathbb{R}}^n⋊O(n)$

Isometries of ${\mathbb{E}}^n$ are the affine maps $\mathbf{x}\mapsto A\mathbf{x}+\mathbf{b}$ with $A\in O(n)$ (derived for $n=2$ here). The group is the semidirect product

$$E(n)={\mathbb{R}}^n⋊O(n),$$

with the translations ${\mathbb{R}}^n$ a normal subgroup acted on by the rotations/reflections $O(n)$. The existence of a normal translation subgroup is the group-theoretic signature of flatness, $K=0$.

2.2 Spherical: $O(n+1)$

Isometries of $S^n\subset {\mathbb{R}}^{n+1}$ are the linear maps preserving the Euclidean form and the sphere — the orthogonal group

$$\mathrm{Isom}(S^n)=O(n+1),{\mathrm{Isom}}^{+}(S^n)=SO(n+1).$$

Unlike the Euclidean case there is no normal translation subgroup: $SO(n+1)$ is (for $n\ge 2$, $n\ne 3$) simple. Positive curvature "curls up" translations into rotations — a sphere has no flat direction to slide along. Passing to elliptic geometry ${\mathbb{RP}}^n$ replaces $O(n+1)$ by $O(n+1)/\{\pm I\}$.

2.3 Hyperbolic: $O(n,1)$

Isometries of ${\mathbb{H}}^n$ are realized cleanly on the hyperboloid model: the linear maps of ${\mathbb{R}}^{n,1}$ preserving the Minkowski form $⟨u,v⟩=u_0{v}_0-\sum u_i{v}_i$ and the upper sheet. This is the Lorentz group

$$\mathrm{Isom}({\mathbb{H}}^n)=O^{+}(n,1)(\text{upper-sheet-preserving part of }O(n,1)).$$

For $n=2$ there is the exceptional isomorphism

$${\mathrm{Isom}}^{+}({\mathbb{H}}^2)\cong PSL(2,\mathbb{R}),$$

acting on the upper half-plane by real Möbius transformations $z\mapsto \frac{az+b}{cz+d}$, $ad-bc=1$. Negative curvature again admits no flat translations; instead its orientation-preserving isometries are classified as elliptic (one fixed point inside — rotations), parabolic (one fixed point on the boundary — horocycle shifts), or hyperbolic (two boundary fixed points — translations along a geodesic axis), by the trace of the matrix.

3. Homogeneous Spaces: $M=G/H$

Because each $G=\mathrm{Isom}(M)$ acts transitively (homogeneity), we can reconstruct the space from the group. Fix a basepoint $o\in M$ and let $H=\mathrm{Stab}(o)$ be its stabilizer (the isometries fixing $o$ — the "rotations about $o$"). Then

$$M\cong G/H,$$

the space of cosets, with $H$ the isotropy group. Reading off:

In every case the isotropy group is the same $O(n)$ — the rotations of the tangent space — expressing isotropy: at any point the geometry looks rotationally symmetric. The three geometries differ only in the "translational" part $G/H$ that moves the basepoint around. These are the three Riemannian symmetric spaces of constant curvature; the $G/H$ presentation is the entry point to the general theory (Helgason) and the template for the Erlangen program, which promotes "geometry $=$ a group acting on a space" to a definition.

4. The Bridge to Physics: $O(n,1)$ is the Lorentz Group

The single most important cross-connection in this spine:

Hyperbolic isometries $=$ Lorentz transformations. The isometry group of ${\mathbb{H}}^n$ is $O^{+}(n,1)$, which is by definition the Lorentz group of $(n+1)$-dimensional Minkowski spacetime — the linear maps preserving the interval $t^2-|\mathbf{x}{|}^2$.

The dictionary is exact and physically loaded:

• The hyperboloid model $t^2-|\mathbf{x}{|}^2=1$ is the mass shell / four-velocity space of special relativity: the set of unit future-timelike vectors. Relativistic velocity space is a hyperbolic space of curvature $-1$.
• A Lorentz boost is a hyperbolic translation along a geodesic of ${\mathbb{H}}^n$; the boost parameter — the rapidity — is precisely hyperbolic arc length, which is why rapidities add while velocities do not.
• The failure of boosts to commute (producing the Thomas–Wigner rotation) is the holonomy of the negatively-curved velocity space: transporting a frame around a triangle of boosts rotates it by the triangle's hyperbolic defect.

Adjoining spacetime translations to $O(n,1)$ gives the Poincaré group ${\mathbb{R}}^{n,1}⋊O(n,1)$ — the exact Minkowski analogue of the Euclidean $E(n)={\mathbb{R}}^n⋊O(n)$, with the indefinite metric replacing the definite one. This parallel is developed on the physics side in SR/group/lorentz-poincare.md and, for the Lie-algebra structure and representation theory, in group-theory/README.md.

5. Discrete Subgroups and Quotient Geometries

Restricting to discrete subgroups $\Gamma \subset G$ acting freely gives the non-simply-connected constant-curvature spaces of the Killing–Hopf theorem:

• $\Gamma \subset E(n)$: the crystallographic groups — wallpaper and space groups — whose quotients are flat tori and their relatives.
• $\Gamma \subset O(n+1)$: finite groups giving spherical space forms (lens spaces, ${\mathbb{RP}}^n$).
• $\Gamma \subset PSL(2,\mathbb{R})$: Fuchsian groups, whose quotients ${\mathbb{H}}^2/\Gamma$ are the hyperbolic surfaces — every closed surface of genus $\ge 2$ — central to Riemann surfaces, modular forms, and Teichmüller theory.

Thus the isometry group organizes not only the three geometries but all their quotients, and the study of $\Gamma$ inside $G$ is where geometry meets group theory and number theory.

6. Summary

The isometry group turns each geometry into an algebraic object:

$${\mathbb{E}}^n\leftrightarrow {\mathbb{R}}^n⋊O(n),S^n\leftrightarrow O(n+1),{\mathbb{H}}^n\leftrightarrow O(n,1),$$

each a homogeneous space $G/O(n)$ differing only in the sign of the curvature and the signature of the invariant form (definite $\to$ sphere, degenerate/affine $\to$ flat, indefinite $\to$ hyperbolic). This is the perspective that the Erlangen program elevates to the definition of geometry, and the bridge by which hyperbolic geometry became the mathematical home of special relativity.