Principal homogeneous space (G-torsor)

Let Lie group $G$ act on a nonempty set $X$ on the left.

A principal homogeneous space for $G$ (also called a $G$-torsor) is a set $X$ together with a left action $G \times X \to X$, $(g,x)\mapsto g\cdot x$, such that:

1. (Free) If $g\cdot x = x$ for some $x\in X$, then $g=e$.
2. (Transitive) For any $x,y\in X$ there exists $g\in G$ with $g\cdot x = y$.

Equivalently, fixing any $x_0\in X$, the orbit map

is a bijection; different choices of $x_0$ differ by right multiplication in $G$, so there is no canonical identification of $X$ with $G$ unless a basepoint is chosen.

If $X$ is a smooth manifold and the action is smooth , then $\theta_{x_0}$ is a diffeomorphism $G\cong X$ for each choice of $x_0$.

A basic source of torsors in geometry: if $P\to M$ is a principal G-bundle , then each fiber $P_x$ is a (right) $G$-torsor under the bundle’s right action.

Examples

1. The group acting on itself. $G$ is a $G$-torsor under left translation: $g\cdot h = gh$.
2. Affine spaces. Any affine space $A$ modeled on a vector space $V$ is a torsor for the additive Lie group $(V,+)$ via translations $v\cdot a := a+v$.
3. Fibers of a principal bundle. For a principal bundle $\pi:P\to M$, each fiber $\pi^{-1}(x)$ is a torsor for $G$: for any $p,q\in \pi^{-1}(x)$ there is a unique $g\in G$ with $p\cdot g=q$.