6.1 Isometries and Riemannian isometry groups

Isometries are transformations that preserve distances and angles in geometric spaces. They're crucial for understanding symmetries in Riemannian manifolds, including rotations, translations, and reflections in various geometries like Euclidean, hyperbolic, and spherical.

The Riemannian isometry group consists of all isometries of a manifold, forming a Lie group with specific properties. This group's structure reveals important information about the manifold's geometry, including its curvature, symmetries, and topological features.

Definition of isometries

• Isometries are transformations that preserve the metric structure of a space, maintaining distances and angles between points
• In the context of Metric Differential Geometry, isometries play a crucial role in understanding the symmetries and properties of Riemannian manifolds

Preserving metric properties

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• Isometries preserve the length of curves and the angle between tangent vectors at each point
• For any two points $p$ and $q$ in a metric space $(M, d)$, an isometry $f: M \rightarrow M$ satisfies $d(f(p), f(q)) = d(p, q)$
• Isometries also preserve other metric properties such as curvature and volume

Isometries as distance-preserving maps

• Isometries can be characterized as distance-preserving maps between metric spaces
• Given two metric spaces $(M_1, d_1)$ and $(M_2, d_2)$, a map $f: M_1 \rightarrow M_2$ is an isometry if $d_2(f(p), f(q)) = d_1(p, q)$ for all $p, q \in M_1$
• Isometries are always injective (one-to-one) and continuous

Local vs global isometries

• Local isometries preserve the metric structure in a neighborhood of each point, but may not be globally distance-preserving
• Global isometries, also known as metric isometries, preserve distances between all pairs of points in the entire space
• Every global isometry is a local isometry, but the converse is not always true (consider the map $f(x) = x$ from $\mathbb{R}$ to $\mathbb{R}_{\geq 0}$)

Examples of isometries

Euclidean isometries

• Isometries of Euclidean space $\mathbb{R}^n$ include translations, rotations, and reflections
• Translations shift all points by a fixed vector, preserving distances and angles (parallel transport)
• Rotations are linear transformations that preserve the origin and the length of vectors (orthogonal matrices)
• Reflections flip points across a hyperplane, preserving distances but reversing orientation

Hyperbolic isometries

• Isometries of the hyperbolic plane $\mathbb{H}^2$ and hyperbolic space $\mathbb{H}^n$ include translations along geodesics, rotations around points, and reflections across geodesics
• Hyperbolic translations shift points along a geodesic by a fixed hyperbolic distance, preserving the distance between any two points on the geodesic
• Hyperbolic rotations rotate points around a fixed point by a given angle, preserving distances and angles
• Hyperbolic reflections flip points across a geodesic, preserving distances but reversing orientation

Isometries on surfaces

• Isometries of surfaces, such as the sphere $\mathbb{S}^2$ or the torus $\mathbb{T}^2$, can be classified based on their fixed points and the topology of the surface
• On the sphere, isometries include rotations around any axis passing through the center and reflections across any great circle
• On the torus, isometries include translations along the two fundamental loops and rotations by multiples of $\pi/2$ around certain points
• Isometries on surfaces can be studied using the concept of covering spaces and the action of the fundamental group on the universal cover

Properties of isometries

Isometries as diffeomorphisms

• Isometries between smooth Riemannian manifolds are always smooth diffeomorphisms
• The differential of an isometry at each point is an orthogonal linear transformation, preserving the inner product on tangent spaces
• The inverse of an isometry is also an isometry, making the set of isometries a group under composition

Isometries and geodesics

• Isometries map geodesics to geodesics, preserving the length and minimizing properties of geodesics
• If $\gamma: [a, b] \rightarrow M$ is a geodesic in a Riemannian manifold $M$ and $f: M \rightarrow M$ is an isometry, then $f \circ \gamma$ is also a geodesic
• Isometries preserve the Levi-Civita connection and the curvature tensor of a Riemannian manifold

Fixed points and invariant sets

• A point $p \in M$ is a fixed point of an isometry $f: M \rightarrow M$ if $f(p) = p$
• The set of fixed points of an isometry can be empty, a single point, or a submanifold of $M$
• An invariant set under an isometry $f$ is a subset $A \subseteq M$ such that $f(A) = A$
• The fixed point set and invariant sets of an isometry provide information about its geometric and topological properties

Riemannian isometry groups

Definition and structure

• The set of all isometries of a Riemannian manifold $(M, g)$ forms a group under composition, called the isometry group and denoted by $\mathrm{Isom}(M, g)$
• The isometry group is a subgroup of the diffeomorphism group $\mathrm{Diff}(M)$ and inherits a natural topology from it
• The isometry group can be finite, discrete, or continuous, depending on the geometry and topology of the manifold

Lie group properties

• For connected Riemannian manifolds, the isometry group is a Lie group, with a smooth group operation and a compatible smooth structure
• The Lie algebra of the isometry group consists of Killing vector fields, which are vector fields that generate one-parameter subgroups of isometries
• The dimension of the isometry group is bounded by $\frac{n(n+1)}{2}$, where $n$ is the dimension of the manifold, with equality achieved only for spaces of constant curvature

Orbits and isotropy subgroups

• The orbit of a point $p \in M$ under the action of the isometry group is the set $\mathrm{Isom}(M, g) \cdot p = \{f(p) \mid f \in \mathrm{Isom}(M, g)\}$
• Orbits partition the manifold into equivalence classes of points related by isometries
• The isotropy subgroup (or stabilizer) of a point $p$ is the subgroup of isometries that fix $p$: $\mathrm{Isom}_p(M, g) = \{f \in \mathrm{Isom}(M, g) \mid f(p) = p\}$
• The orbit of $p$ is diffeomorphic to the quotient space $\mathrm{Isom}(M, g) / \mathrm{Isom}_p(M, g)$

Classification of isometries

Rotations and translations

• Rotations are isometries that fix a point (in Euclidean and hyperbolic spaces) or a geodesic (in spherical geometry) and rotate the space around it
• Translations are isometries that move all points by a fixed distance in a given direction, without any fixed points
• In Euclidean space, rotations and translations generate the entire isometry group, while in hyperbolic and spherical geometries, they form proper subgroups

Reflections and glide reflections

• Reflections are isometries that flip the space across a hyperplane (in Euclidean space) or a geodesic (in hyperbolic and spherical geometries)
• Glide reflections are compositions of a reflection and a translation in the direction parallel to the reflecting hyperplane or geodesic
• Reflections and glide reflections are orientation-reversing isometries, while rotations and translations are orientation-preserving

Discrete vs continuous isometry groups

• Discrete isometry groups have a discrete topology and are typically generated by a finite set of isometries, such as reflections or rotations of a specific angle
• Continuous isometry groups have a continuous topology and often arise from the action of a Lie group on the manifold
• Examples of discrete isometry groups include the symmetry groups of regular polyhedra and the fundamental groups of compact hyperbolic surfaces
• Examples of continuous isometry groups include the orthogonal group $\mathrm{O}(n)$ acting on Euclidean space and the Lorentz group $\mathrm{O}(1, n-1)$ acting on Minkowski spacetime

Isometry groups of model spaces

Isometry group of Euclidean space

• The isometry group of Euclidean space $\mathbb{R}^n$ is the Euclidean group $\mathrm{E}(n)$, which is the semi-direct product of the orthogonal group $\mathrm{O}(n)$ and the translation group $\mathbb{R}^n$
• The orthogonal group $\mathrm{O}(n)$ consists of all linear isometries (rotations and reflections) of $\mathbb{R}^n$, while the translation group $\mathbb{R}^n$ acts by adding a constant vector to each point
• The dimension of the Euclidean group is $\frac{n(n+1)}{2}$, which is the maximum possible dimension for the isometry group of an $n$-dimensional Riemannian manifold

Isometry groups of spheres

• The isometry group of the $n$-dimensional sphere $\mathbb{S}^n$ is the orthogonal group $\mathrm{O}(n+1)$, acting on the sphere as a submanifold of $\mathbb{R}^{n+1}$
• The orientation-preserving isometries of $\mathbb{S}^n$ form the special orthogonal group $\mathrm{SO}(n+1)$, which is the identity component of $\mathrm{O}(n+1)$
• The isometry group of $\mathbb{S}^n$ acts transitively on the sphere, meaning that any two points can be mapped to each other by an isometry

Isometry groups of hyperbolic spaces

• The isometry group of the $n$-dimensional hyperbolic space $\mathbb{H}^n$ is the orthogonal group $\mathrm{O}(1, n)$ of the Minkowski space $\mathbb{R}^{1,n}$, acting on $\mathbb{H}^n$ as a submanifold
• The orientation-preserving isometries of $\mathbb{H}^n$ form the special orthogonal group $\mathrm{SO}(1, n)$, which is the identity component of $\mathrm{O}(1, n)$
• The isometry group of $\mathbb{H}^n$ acts transitively on the space and preserves the ideal boundary $\partial \mathbb{H}^n$, which is a sphere of dimension $n-1$

Applications of isometries

Symmetries in geometry and physics

• Isometries are used to study symmetries in geometry, such as the symmetries of regular polyhedra, lattices, and tilings
• In physics, isometries of spacetime (Poincaré group) and internal spaces (gauge groups) play a fundamental role in the formulation of theories such as special and general relativity, quantum mechanics, and quantum field theory
• Noether's theorem relates continuous symmetries (isometries) to conservation laws, such as the conservation of energy, momentum, and charge

Isometric embeddings and rigidity

• An isometric embedding is a map between Riemannian manifolds that preserves the metric tensor
• The Nash embedding theorem states that every compact Riemannian manifold can be isometrically embedded into a Euclidean space of sufficiently high dimension
• Rigidity theorems, such as the Mostow rigidity theorem and the Margulis superrigidity theorem, study the existence and uniqueness of isometric embeddings or maps between certain classes of manifolds

Isospectral manifolds and inverse problems

• Two Riemannian manifolds are called isospectral if they have the same spectrum of the Laplace-Beltrami operator
• Isospectral manifolds share many geometric and topological properties, but may not be isometric
• The study of isospectral manifolds is related to inverse spectral problems, which aim to determine the geometry of a manifold from its spectrum
• Isometries play a role in the construction and classification of isospectral manifolds, as they preserve the spectrum of the Laplace-Beltrami operator