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

Let $G$ be a Lie group. In this convention, a principal $G$ -bundle is a fiber bundle $\pi:P\to M$ equipped with a right smooth action

R: P\times G\to P,\qquad (p,g)\mapsto p\cdot g,

which is free and transitive on each fiber, with $M\cong P/G$ .

Consequences for formulas

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 $g\in G$ , the convention is

(R_g)^*\omega=\mathrm{Ad}(g^{-1})\,\omega,

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

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

Transition functions from local sections. If $s_i,s_j:U\to P$ are local sections, the transition map $g_{ij}:U\to G$ is defined by $s_j=s_i\cdot g_{ij}$ .

Gauge transformations. A gauge transformation $\Phi:P\to P$ satisfies $\Phi(p\cdot g)=\Phi(p)\cdot g$ .