Isometries are transformations that preserve distances and angles in geometric spaces. They're crucial for understanding symmetries in Riemannian manifolds, including , , and in various geometries like Euclidean, hyperbolic, and spherical.
The Riemannian 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
Top images from around the web for Preserving metric properties
differential geometry - Riemannian metrics and how spaces look - Mathematics Stack Exchange View original
Is this image relevant?
1 of 3
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 f:M→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 (M1,d1) and (M2,d2), a map f:M1→M2 is an isometry if d2(f(p),f(q))=d1(p,q) for all p,q∈M1
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 is a , but the converse is not always true (consider the map f(x)=x from R to R≥0)
Examples of isometries
Euclidean isometries
Isometries of Euclidean space Rn 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 H2 and hyperbolic space Hn include translations along geodesics, rotations around points, and reflections across geodesics
Hyperbolic translations shift points along a 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 S2 or the torus T2, 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 π/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 γ:[a,b]→M is a geodesic in a Riemannian manifold M and f:M→M is an isometry, then f∘γ is also a geodesic
Isometries preserve the Levi-Civita connection and the of a Riemannian manifold
Fixed points and invariant sets
A point p∈M is a fixed point of an isometry f:M→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 under an isometry f is a subset A⊆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 Isom(M,g)
The isometry group is a subgroup of the diffeomorphism group 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 , which are vector fields that generate one-parameter subgroups of isometries
The dimension of the isometry group is bounded by 2n(n+1), where n is the dimension of the manifold, with equality achieved only for spaces of constant curvature
Orbits and isotropy subgroups
The of a point p∈M under the action of the isometry group is the set Isom(M,g)⋅p={f(p)∣f∈Isom(M,g)}
Orbits partition the manifold into equivalence classes of points related by isometries
The (or stabilizer) of a point p is the subgroup of isometries that fix p: Isomp(M,g)={f∈Isom(M,g)∣f(p)=p}
The orbit of p is diffeomorphic to the quotient space Isom(M,g)/Isomp(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 O(n) acting on Euclidean space and the Lorentz group O(1,n−1) acting on Minkowski spacetime
Isometry groups of model spaces
Isometry group of Euclidean space
The isometry group of Euclidean space Rn is the E(n), which is the semi-direct product of the orthogonal group O(n) and the translation group Rn
The orthogonal group O(n) consists of all linear isometries (rotations and reflections) of Rn, while the translation group Rn acts by adding a constant vector to each point
The dimension of the Euclidean group is 2n(n+1), 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 Sn is the orthogonal group O(n+1), acting on the sphere as a submanifold of Rn+1
The orientation-preserving isometries of Sn form the SO(n+1), which is the identity component of O(n+1)
The isometry group of Sn 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 Hn is the orthogonal group O(1,n) of the Minkowski space R1,n, acting on Hn as a submanifold
The orientation-preserving isometries of Hn form the special orthogonal group SO(1,n), which is the identity component of O(1,n)
The isometry group of Hn acts transitively on the space and preserves the ideal boundary ∂Hn, 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 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