Convention: principal bundles use a right G-action on P

Let $G$ be a Lie group .

Convention

A principal G-bundle is a fiber bundle $\pi:P\to M$ equipped with a right smooth action

which is free and fiberwise transitive, with $M\cong P/G$.

All equivariance conditions are written with respect to this right action. In particular, for a principal connection 1-form $\omega\in\Omega^1(P;\mathfrak g)$ and for $g\in G$, we use

and the induced vertical identification uses fundamental vector fields defined from the right action (see fundamental vector field convention ).

This convention fixes the sign choices appearing in curvature and covariant differentiation identities.

Examples

1. Frame bundle. For a rank-$n$ vector bundle $E\to M$, the bundle of frames $P=\mathrm{Fr}(E)$ carries a natural right $GL(n,\mathbb R)$-action by postcomposition of frames, $(u,g)\mapsto u\circ g$.

2. Transition functions from local sections. With right actions, if $s_i,s_j:U\to P$ are local sections, the transition map $g_{ij}:U\to G$ is defined by $s_j=s_i\cdot g_{ij}$ (see transition functions from local sections ).

3. Gauge transformations. A gauge transformation is typically a $G$-equivariant diffeomorphism $\Phi:P\to P$ commuting with the right action; writing the action on the right makes $\Phi(p\cdot g)=\Phi(p)\cdot g$ the natural equivariance condition.