Isometries and Conformal Maps

Isometries are the structure-preserving maps of Riemannian geometry, while conformal maps preserve angles but may distort distances. Both play central roles in understanding the rigidity and flexibility of geometric structures.

Isometries

Definition6.4Isometry

A smooth map $f: (M, g) \to (N, h)$ between Riemannian manifolds is an isometry if $f^*h = g$ , i.e., $h_{f(p)}(df_p(u), df_p(v)) = g_p(u, v)$ for all $p \in M$ and $u, v \in T_pM$ . An isometry is necessarily a diffeomorphism onto its image. A local isometry satisfies the same condition but need not be globally injective.

ExampleClassical isometries

Euclidean isometries: Translations, rotations, and reflections form the isometry group $\mathrm{Isom}(\mathbb{R}^n) = O(n) \ltimes \mathbb{R}^n$ .
Spherical isometries: $\mathrm{Isom}(S^n) = O(n+1)$ , acting by orthogonal transformations of $\mathbb{R}^{n+1}$ .
Hyperbolic isometries: $\mathrm{Isom}(\mathbb{H}^n) = O^+(1, n)$ , the orthochronous Lorentz group (for the hyperboloid model).
Flat torus: The torus $T^2 = \mathbb{R}^2/\mathbb{Z}^2$ has a flat metric locally isometric to $\mathbb{R}^2$ , but not globally isometric.

Theorem6.2Myers-Steenrod theorem

The isometry group $\mathrm{Isom}(M, g)$ of a Riemannian manifold is a Lie group with respect to the compact-open topology. Its dimension satisfies $\dim \mathrm{Isom}(M) \leq \frac{n(n+1)}{2}$ , with equality if and only if $(M, g)$ has constant sectional curvature.

Conformal Maps

Definition6.5Conformal map

A smooth map $f: (M, g) \to (N, h)$ is conformal if $f^*h = e^{2\varphi} g$ for some smooth function $\varphi: M \to \mathbb{R}$ . Equivalently, $f$ preserves angles between tangent vectors but may scale lengths. Two metrics $g$ and $\tilde{g} = e^{2\varphi} g$ are conformally equivalent.

ExampleConformal examples

Stereographic projection $S^n \setminus \{N\} \to \mathbb{R}^n$ is conformal but not an isometry.
Mercator projection of the sphere to the cylinder is conformal, which is why it preserves angles on maps (but distorts areas near the poles).
On Riemann surfaces ( $n = 2$ ), conformal maps are holomorphic or anti-holomorphic. The group of conformal automorphisms of $\hat{\mathbb{C}}$ is $\mathrm{PSL}_2(\mathbb{C})$ (Mobius transformations).

Conformal Geometry in Dimension 2

RemarkConformal classes in dimension 2

In dimension 2, the conformal class of a Riemannian metric is equivalent to a complex structure (by the uniformization theorem). Every Riemannian surface is conformally equivalent to one of: $S^2$ (curvature $> 0$ ), $\mathbb{R}^2$ or $T^2$ (curvature $= 0$ ), or a hyperbolic surface $\mathbb{H}^2/\Gamma$ (curvature $< 0$ ). The space of conformal structures on a surface of genus $g \geq 2$ is the Teichmuller space, a $(6g - 6)$ -dimensional manifold.

Theorem6.3Liouville's theorem (conformal rigidity)

For $n \geq 3$ , every conformal diffeomorphism of an open subset of $\mathbb{R}^n$ (or $S^n$ ) is the restriction of a Mobius transformation: a composition of translations, rotations, dilations, and inversions. In particular, the conformal group of $S^n$ is the $(n+1)(n+2)/2$ -dimensional Lie group $\mathrm{SO}^+(1, n+1)$ .

RemarkContrast with dimension 2

Liouville's theorem shows that conformal geometry is rigid in dimensions $\geq 3$ : the conformal group is finite-dimensional. In dimension 2, however, conformal maps are holomorphic functions, giving an infinite-dimensional group. This dichotomy profoundly influences geometry and physics.