Difference of two principal connections is tensorial

Let $\pi:P\to M$ be a principal G-bundle with Lie group $G$ and Lie algebra $\mathfrak g$ (see Lie algebra of a Lie group ). Let $\omega,\omega'\in \Omega^1(P;\mathfrak g)$ be connection $1$-forms of two principal connections (see connection 1-form on a principal bundle ).

Define their difference

Lemma

The $1$-form $a$ is tensorial of type $\mathrm{Ad}$, meaning:

1. (Horizontal) $a_p(v)=0$ for every vertical vector $v\in V_pP$ (equivalently $\iota_{X^\#}a=0$ for all fundamental vertical fields $X^\#$).
2. ($\mathrm{Ad}$-equivariant) $(R_g)^*a = \mathrm{Ad}(g^{-1})\,a$ for all $g\in G$ (see adjoint action ).

Consequently, $a$ descends to a well-defined $1$-form on the base with values in the adjoint bundle:

where $\mathrm{ad}(P)=P\times_{\mathrm{Ad}}\mathfrak g$ is the adjoint bundle (compare construction of the adjoint Lie algebra bundle and sections of ad(P) ).

This tensoriality fact underlies many standard constructions, including transgression formulas (see transgression forms ) and the description of the affine space of connections (compare bundle of connections ).

Examples

1. Trivial bundle: local gauge potentials differ by a basic form.
   On the trivial principal bundle $P=M\times G$ (see trivial principal bundle ), a principal connection corresponds to a $\mathfrak g$-valued $1$-form $A\in\Omega^1(M;\mathfrak g)$ via a global section. If $\omega,\omega'$ correspond to $A,A'$, then

   $$a=\omega'-\omega \quad \text{corresponds to} \quad A'-A\in\Omega^1(M;\mathfrak g),$$

   which is a genuine $1$-form on $M$ (hence automatically horizontal/basic after pullback to $P$).

2. U(1) bundles: two connections differ by an ordinary real 1-form.
   For $G=U(1)$, the adjoint action is trivial, so $\mathrm{ad}(P)\cong M\times i\mathbb R$. Thus the difference of two $U(1)$-connections is simply an $i\mathbb R$-valued $1$-form on $M$. Concretely, if $A$ and $A'$ are local connection $1$-forms (see local connection 1-form ), then $A'-A$ patches globally as an $i\mathbb R$-valued form.

3. Frame bundle: change of linear connection is tensorial.
   Let $E\to M$ be a rank-$n$ vector bundle and let $P=\mathrm{Fr}(E)$ be its frame bundle (see frame bundle of a vector bundle ). Two vector bundle connections on $E$ correspond to two principal connections on $P$ (compare connections via frame bundles ). Their difference is a tensorial $1$-form on $P$, which corresponds to a $1$-form on $M$ with values in $\mathrm{End}(E)$, i.e. the usual “difference tensor” between linear connections.