Convention: fundamental vector field uses the right action

Let $\pi:P\to M$ be a principal G-bundle with the right action $(p,g)\mapsto p\cdot g$ as in the right-action convention . Let $\mathfrak g$ be the Lie algebra of $G$.

Convention

For $X\in\mathfrak g$, the fundamental vector field (also called the infinitesimal generator) is the vector field $X^\#\in\Gamma(TP)$ defined by

With this convention:

• $X^\#$ is vertical and $\pi_*X^\#=0$.
• The connection 1-form $\omega$ satisfies $\omega(X^\#)=X$.
• Right translation behaves by $(R_g)_*X^\# = (\mathrm{Ad}(g^{-1})X)^\#$.

This fixes sign conventions in identities involving Lie brackets of fundamental fields and in the structure equations for connections and curvature.

Examples

1. Trivial bundle. If $P=M\times G$ with right action $(x,h)\cdot g=(x,hg)$, then at $(x,h)$ one has

   $$X^\#_{(x,h)}=(0,\ (R_h)_*X),$$

   i.e. the fundamental field points purely along the $G$-factor.

2. Right-invariant fields on $G$. On $P=G$ viewed as a principal $G$-bundle over a point with right action by right multiplication, $X^\#$ is precisely the right-invariant vector field generated by $X$.

3. Sign comparison with a left action. If one instead defined fundamental fields using a left action $L_{\exp(tX)}$, the resulting infinitesimal generator differs by a sign in formulas relating brackets and the adjoint action; the present convention avoids that mismatch for principal bundles written with right actions.