Tensorial forms and ad(P)-valued forms

Let $\pi:P\to M$ be a principal G-bundle over a smooth manifold $M$, where $G$ is a Lie group with Lie algebra $\mathfrak g$.

The adjoint bundle of $P$ is the associated vector bundle

where $G$ acts on $\mathfrak g$ by the adjoint representation $\operatorname{Ad}$.

A $\mathfrak g$-valued differential k-form $\omega\in\Omega^k(P;\mathfrak g)$ is called tensorial of type $\operatorname{Ad}$ if it satisfies:

1. Horizontality: $\omega_p(v_1,\dots,v_k)=0$ whenever at least one $v_i\in\ker(d\pi_p)$ (i.e., a vertical tangent vector).
2. Equivariance: for every $g\in G$, $(R_g)^*\omega = \operatorname{Ad}(g^{-1})\,\omega,$ where $R_g:P\to P$ is the right action.

Write $\Omega^k_{\mathrm{tens}}(P;\mathfrak g)$ for the vector space of such tensorial forms.

Theorem (TFAE: tensorial forms vs ad(P)-valued forms)

There is a natural vector space isomorphism

More explicitly:

• Given $\omega\in\Omega^k_{\mathrm{tens}}(P;\mathfrak g)$, define $\alpha\in\Omega^k(M;\operatorname{ad}(P))$ by

  $$\alpha_x(u_1,\dots,u_k)\;:=\;\bigl[p,\;\omega_p(\tilde u_1,\dots,\tilde u_k)\bigr]\in \operatorname{ad}(P)_x,$$

  where $p\in P$ satisfies $\pi(p)=x$, and $\tilde u_i\in T_pP$ are any tangent vectors with $d\pi_p(\tilde u_i)=u_i$. This is well-defined because vertical components do not affect $\omega$ (horizontality) and changing $p$ in the fiber changes $\omega$ by $\operatorname{Ad}(g^{-1})$ (equivariance), exactly matching the associated-bundle identification.

• Conversely, given $\alpha\in\Omega^k(M;\operatorname{ad}(P))$, define $\omega\in\Omega^k(P;\mathfrak g)$ by requiring that for $p\in P$ and $v_i\in T_pP$,

  $$\alpha_{\pi(p)}(d\pi_p v_1,\dots,d\pi_p v_k)=\bigl[p,\;\omega_p(v_1,\dots,v_k)\bigr]\in\operatorname{ad}(P)_{\pi(p)}.$$

  This forces $\omega$ to be horizontal, and the associated-bundle relation forces $\operatorname{Ad}$-equivariance.

These constructions are inverse to each other and are natural with respect to bundle morphisms covering the identity on $M$.

A key application is that the curvature of a principal connection is tensorial of type $\operatorname{Ad}$, hence canonically corresponds to an $\operatorname{ad}(P)$-valued 2-form on the base.

Examples

1. Curvature descends to the base.
   If $\Theta$ is a principal connection on $P$ with curvature $F_\Theta\in\Omega^2(P;\mathfrak g)$, then $F_\Theta$ is horizontal and $\operatorname{Ad}$-equivariant. The theorem identifies $F_\Theta$ with a unique element of $\Omega^2(M;\operatorname{ad}(P))$.

2. Trivial bundle case.
   On the trivial principal bundle $P=M\times G$, the adjoint bundle is canonically isomorphic to $M\times\mathfrak g$. Under this identification, a tensorial $\omega$ corresponds to an ordinary $\mathfrak g$-valued form on $M$ (written in a chosen global section), and the correspondence is given by pullback along the section and associated-bundle identification.

3. Scalar “basic” forms as a special case.
   If one replaces $\mathfrak g$ by a trivial $G$-representation (so equivariance becomes $G$-invariance), then horizontal $G$-invariant forms on $P$ correspond exactly to ordinary differential forms on $M$ via pullback along $\pi$.