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

\operatorname{ad}(P):=P\times_{\operatorname{Ad}}\mathfrak g \;\longrightarrow\; M,

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

For $k\ge0$ , a $\mathfrak g$ -valued differential k-form $\omega\in\Omega^k(P;\mathfrak g)$ is called tensorial of type $\operatorname{Ad}$ if it satisfies:

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).
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.

There is a natural vector space isomorphism

\Omega^k_{\mathrm{tens}}(P;\mathfrak g)\;\cong\;\Omega^k\!\bigl(M;\operatorname{ad}(P)\bigr).

Explicit correspondence

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

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 $\Omega^2(M;\operatorname{ad}(P))$ .

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.

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 $\pi$ .