Principal Homogeneous Space

Let $G$ be a Lie group . A principal homogeneous space (or $G$-torsor) is a smooth manifold $P$ equipped with a smooth action

that is:

• Free: if $g\cdot p=p$ for some $p\in P$, then $g=e$.
• Transitive: for any $p,q\in P$, there exists $g\in G$ with $g\cdot p=q$.

Equivalently, the action is simply transitive: for each $p,q\in P$ there is a unique $g\in G$ with $g\cdot p=q$.

Choosing a basepoint identifies it with the group

Fix $p_0\in P$. The map

is a diffeomorphism. This identifies $P$ with $G$, but the identification depends on the choice of $p_0$, so there is generally no canonical “origin” in a torsor.

Examples and related notions

• $G$ acting on itself by left translation is a principal homogeneous space.
• More generally, a transitive action with stabilizer $H$ gives a homogeneous space $G/H$; compare quotients . A principal homogeneous space is the special case $H=\{e\}$.

Principal homogeneous spaces are the geometric backdrop for principal bundles and symmetry without a preferred identity element.