Connection 1-form on a principal bundle

Definition of a principal connection 1-form and the horizontal distribution it determines.

Connection 1-form on a principal bundle

Let $\pi:P\to M$ be a principal G-bundle (with the standard right action convention ), and let $\mathfrak g$ be the Lie algebra of $G$ .

A (principal) connection 1-form on $P$ is a $\mathfrak g$ -valued 1-form

\omega \in \Omega^1(P;\mathfrak g)

such that:

Reproduction on verticals. For every $X\in\mathfrak g$ , if $X^\#$ denotes the fundamental vector field on $P$ generated by $X$ (so $X^\#$ is vertical), then
$\omega(X^\#)=X .$
(This is the reproduction property .)
Equivariance. For every $g\in G$ ,
$(R_g)^*\omega = \mathrm{Ad}_{g^{-1}}\omega,$
where $\mathrm{Ad}$ is the adjoint action of $G$ on $\mathfrak g$ .

These two axioms are equivalent to specifying a principal connection : the horizontal space at $p\in P$ is

H_p := \ker(\omega_p)\subset T_pP,

giving a smooth $G$ -invariant horizontal distribution complementary to the vertical subbundle .

If $s:U\to P$ is a smooth local section (as in constructing a local trivialization from a local section ), then the pullback

A := s^*\omega \in \Omega^1(U;\mathfrak g)

is the corresponding local connection 1-form (“gauge potential”) on $U$ .

From $\omega$ one defines the curvature 2-form

\Omega := d\omega + \tfrac12[\omega\wedge \omega] \in \Omega^2(P;\mathfrak g),

which is horizontal and equivariant, hence descends to local curvature forms on $M$ .

Examples

The bundle $G\to \mathrm{pt}$ . View $P=G$ as a principal $G$ -bundle over a point with right multiplication. The left Maurer–Cartan form $\theta_L$ is a connection 1-form: it reproduces generators of the right action and satisfies the required equivariance under right translation.
Trivial bundle with a chosen potential. For $P=M\times G$ , choose any $A\in\Omega^1(M;\mathfrak g)$ . Writing $(x,g)\in M\times G$ , define
$\omega := \mathrm{Ad}_{g^{-1}}(\pi_M^*A) + \theta_L ,$
where $\theta_L$ is pulled back from the $G$ -factor and $\pi_M:P\to M$ is projection. With respect to the global section $s(x)=(x,e)$ , the local form is exactly $s^*\omega=A$ . When $A=0$ this gives the standard flat connection .
Hopf fibration (abelian case). On the Hopf bundle $S^3\to S^2$ with structure group $U(1)$ , there is a canonical connection 1-form (the Dirac monopole connection ). Since $U(1)$ is abelian, the equivariance condition simplifies, and one can write an explicit $\mathfrak u(1)$ -valued 1-form on $S^3\subset\mathbb C^2$ that annihilates horizontals and evaluates to the generator on the circle fibers.