February 24, 2023
Benefits:
Mathematical language: quotient spaces
An Equivalence Relation \(\sim\) on a set \(X\) is a subset of \(X\times X\) s.t.
An Equivalence Relation \(\sim\) on a set \(X\) is a subset of \(X\times X\) s.t.
The equivalence class of an element \(x\in X\) is
\[[x]:=\{y\in X:x\sim y\}\]
Given an equivalence relation \(\sim\) on \(X\), we can consider the set of equivalence classes \[X/\!\sim\;:=\{[x]:x\in X\}\]
We have the projection map \[\pi:X\to X/\!\sim\quad x\mapsto[x]\]
If \(X\) has certain properties, does \(X/\!\sim\)?
Answer: Sometimes
If \(X\) is a topological space, we can use \(\pi\) to define the quotient topology on \(X/\!\sim\), where \(Y\subseteq X/\!\sim\) is an open set iff \(\pi^{-1}(Y)\) is open in \(X\).
Example of a group: rotations in \(\mathbb{R}^3\), called \(\operatorname{SO}(3)\)
Example of a group action: rotations applied to points in \(\mathbb{R}^3\)
A group action yields an equivalence relation
If \(\mathcal{M}\) is a manifold, and \(G\) is a group that acts on \(\mathcal{M}\), then* \(\mathcal{M}/\!\sim\), also written as \(\mathcal{M}/G\), is a manifold.
*Assuming \(G\) satisfies various technical properties. See Introduction to Smooth Manifolds, Lee, Chapter 21, for details.
Under certain further conditions (which are satisfied by the symmetry groups of rigid bodies), \(\mathcal{M}/G\) gets a Riemannian metric from \(\mathcal{M}\).
Furthermore, the distance function takes the following form: \[d([x],[y])=\inf\{d(x,y):x\in[x],y\in[y]\}\]
For a finite group \(G\), this is just a minimum of finitely many entries.
Infinite symmetry groups still have nice properties?