6.6 Isometric group actions

Isometric group actions are powerful tools for understanding symmetries in metric spaces. They preserve distances between points, allowing us to analyze geometric structures based on their symmetry properties. This concept is crucial for classifying spaces and studying their transformations.

By exploring orbits, isotropy subgroups, and quotient spaces, we gain insights into the behavior of these actions. This knowledge has wide-ranging applications in geometry, topology, and physics, helping us uncover fundamental properties of space and symmetry.

Isometric group actions

• Isometric group actions play a crucial role in understanding the symmetries and transformations of metric spaces while preserving distances between points
• The study of isometric group actions allows for the classification and analysis of geometric structures based on their symmetry properties

Definition of isometric group actions

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• An isometric group action is a map $\phi: G \times X \to X$ where $G$ is a group and $X$ is a metric space
  • The map satisfies the following conditions:
    1. $\phi(e, x) = x$ for all $x \in X$, where $e$ is the identity element of $G$
    2. $\phi(g_1, \phi(g_2, x)) = \phi(g_1g_2, x)$ for all $g_1, g_2 \in G$ and $x \in X$
• For each $g \in G$, the map $\phi_g: X \to X$ defined by $\phi_g(x) = \phi(g, x)$ is an isometry of $X$
  • This means that $d(\phi_g(x), \phi_g(y)) = d(x, y)$ for all $x, y \in X$, where $d$ is the metric on $X$

Examples of isometric group actions

• The group of rotations $SO(n)$ acts isometrically on the $n$-dimensional Euclidean space $\mathbb{R}^n$ by matrix multiplication
  • Each rotation preserves distances between points in $\mathbb{R}^n$
• The group of translations acts isometrically on Euclidean space by shifting points by a fixed vector
  • Translations preserve distances between points and maintain the geometric structure of the space

Orbits of isometric group actions

• The orbit of a point $x \in X$ under an isometric group action $\phi$ is the set $\{gx : g \in G\}$, where $gx = \phi(g, x)$
  • Orbits partition the space $X$ into disjoint subsets, each consisting of points that can be mapped to one another by elements of $G$
• The orbit space $X/G$ is the set of all orbits of the action
  • The orbit space provides a way to study the quotient space of $X$ under the equivalence relation induced by the group action

Isotropy subgroups of isometric group actions

• The isotropy subgroup (or stabilizer) of a point $x \in X$ under an isometric group action $\phi$ is the subgroup $G_x = \{g \in G : gx = x\}$
  • The isotropy subgroup consists of all elements of $G$ that fix the point $x$
• The conjugacy class of an isotropy subgroup is an invariant of the group action
  • Conjugate isotropy subgroups correspond to points in the same orbit

Properties of isometric group actions

• The properties of isometric group actions provide insight into the behavior and structure of the action and its effect on the underlying metric space

Proper vs improper actions

• An isometric group action is proper if for every compact subset $K \subset X$, the set $\{g \in G : gK \cap K \neq \emptyset\}$ is compact in $G$
  • Proper actions have well-behaved quotient spaces and orbits
• Improper actions may have orbits that are not closed or have non-Hausdorff quotient spaces

Discrete vs continuous actions

• A discrete isometric group action is one in which the group $G$ is a discrete group (e.g., the integers or a finite group)
  • Discrete actions often lead to orbits that are discrete subsets of the space
• A continuous isometric group action is one in which the group $G$ is a continuous group (e.g., Lie groups like $SO(n)$ or $\mathbb{R}$)
  • Continuous actions can lead to orbits that are continuous submanifolds of the space

Free vs non-free actions

• An isometric group action is free if for every $x \in X$, the isotropy subgroup $G_x$ is trivial (consists only of the identity element)
  • In a free action, no non-identity element of $G$ fixes any point of $X$
• A non-free action has non-trivial isotropy subgroups for some points in $X$
  • Non-free actions may have fixed points or points with non-trivial stabilizers

Quotient spaces of isometric group actions

• Quotient spaces arise naturally when studying isometric group actions, as they allow for the identification of points that are equivalent under the action

Definition of quotient space

• The quotient space $X/G$ of a metric space $X$ under an isometric group action $\phi$ is the set of all orbits of the action
  • Points in the same orbit are considered equivalent in the quotient space
• The quotient map $\pi: X \to X/G$ sends each point $x \in X$ to its orbit $Gx$

Metric on quotient space

• A metric $d_{X/G}$ can be defined on the quotient space $X/G$ by setting $d_{X/G}(Gx, Gy) = \inf\{d(x', y') : x' \in Gx, y' \in Gy\}$
  • This metric measures the distance between orbits as the infimum of distances between points in the orbits
• The quotient map $\pi: X \to X/G$ is a metric submersion, meaning that it preserves distances between points in $X$ that are mapped to different orbits in $X/G$

Properties of quotient spaces

• The topology of the quotient space $X/G$ is determined by the quotient topology, where a subset $U \subset X/G$ is open if and only if its preimage $\pi^{-1}(U)$ is open in $X$
• If the isometric group action is proper and free, the quotient space $X/G$ is a Hausdorff metric space
  • In this case, the quotient map $\pi: X \to X/G$ is a covering map

Applications of isometric group actions

• Isometric group actions have numerous applications across various branches of mathematics and physics, providing a framework for understanding symmetries and invariance properties

Isometric group actions in geometry

• Isometric group actions are used to classify and study geometric structures based on their symmetries
  • For example, the classification of Euclidean space forms relies on the study of isometric group actions on Euclidean space
• The study of isometric group actions on Riemannian manifolds leads to the development of important concepts such as Killing vector fields and homogeneous spaces

Isometric group actions in topology

• Isometric group actions provide a way to construct new topological spaces from existing ones through the process of taking quotients
  • Quotient spaces under isometric group actions often have interesting topological properties and can be used to construct examples of manifolds and orbifolds
• The study of free and proper isometric group actions is closely related to the theory of principal bundles and fiber bundles in topology

Isometric group actions in physics

• Isometric group actions play a fundamental role in the study of symmetries in physical systems
  • The invariance of physical laws under certain groups of transformations (e.g., the Poincaré group in special relativity) is described using isometric group actions
• The study of gauge theories in physics heavily relies on the concept of principal bundles, which are constructed using free and proper isometric group actions on the total space