Reproduction property

Let $\pi:P\to M$ be a principal $G$ -bundle and let $\omega\in \Omega^1(P;\mathfrak{g})$ be a Lie-algebra-valued 1-form. (When $\omega$ is a connection 1-form, this condition is one of its defining axioms.)

For $X\in \mathfrak{g}$ , the fundamental vector field $X^\#$ is the vector field on $P$ defined by

X^\#_p \coloneqq \left.\frac{d}{dt}\right|_{t=0} \bigl(p\cdot \exp(tX)\bigr).

The reproduction property is the requirement that

\omega(X^\#)=X \quad \text{for all } X\in \mathfrak{g}.

This condition says that $\omega$ restricts on each vertical space $V_p=\ker(d\pi_p)$ to the canonical identification $V_p\cong \mathfrak{g}$ coming from the infinitesimal right action.

Examples

Maurer–Cartan form on a Lie group. For the principal bundle $G\to \{\ast\}$ (right action by multiplication), the left Maurer–Cartan form $\theta=g^{-1}dg$ satisfies $\theta(X^\#)=X$ because $g^{-1}\frac{d}{dt}(g\exp(tX))|_{0}=X$ .

Trivial bundle $U\times G$ . With $\omega=\mathrm{Ad}(g^{-1})A + g^{-1}dg$ as in a standard trivialization, a purely vertical vector at $(x,g)$ has the form $(0,(L_g)_*X)$ , and one checks $\omega(0,(L_g)_*X)=X$ .

The abelian case $U(1)$ . Identify $\mathfrak{u}(1)\cong i\mathbb{R}$ . If $\partial_\theta$ denotes the fundamental field for the standard $U(1)$ -action, the reproduction property is $\omega(\partial_\theta)=1$ (after the usual identification of $i\mathbb{R}$ with $\mathbb{R}$ ).