Theorem: Pullback of a principal connection is a principal connection

Let $f:N\to M$ be a smooth map and let $\pi:P\to M$ be a principal G-bundle with a principal connection $\omega$ .

Let $\mathrm{pr}_P:f^*P\to P$ denote the projection from the pullback bundle $f^*P\subset N\times P$ onto the second factor.

There is a unique principal connection $f^*\omega$ on $f^*P\to N$ characterized by either of the equivalent descriptions:

By connection 1-forms: the connection form on $f^*P$ is $(\mathrm{pr}_P)^*\omega\in\Omega^1(f^*P;\mathfrak g)$ .

By horizontal subspaces: for $(n,p)\in f^*P$ , the horizontal subspace is $H_{(n,p)} := \{\,v\in T_{(n,p)}(f^*P)\mid d(\mathrm{pr}_P)(v)\in H_p\,\},$ where $H_p\subset T_pP$ is the horizontal subspace of $\omega$ at $p$ .

In either description, $f^*\omega$ is $G$ -equivariant and reproduces fundamental vector fields, hence is a principal connection.

Its curvature satisfies $\mathrm{pr}_P^*(\Omega)=\Omega_{f^*\omega}$ , i.e. curvature pulls back under $\mathrm{pr}_P$ .

Examples

Restriction of a connection. If $i:Z\hookrightarrow M$ is an embedding, then $i^*\omega$ is the restriction of $\omega$ to the restricted bundle $P|_Z$ .

Pullback along a parametrized curve. If $\gamma:[0,1]\to M$ , then $\gamma^*P\to[0,1]$ carries the pulled-back connection $\gamma^*\omega$ ; horizontal sections of $\gamma^*P$ are precisely horizontal lifts of $\gamma$ in $P$ .

Trivial bundle picture. For $P=M\times G$ with connection form determined by a $\mathfrak g$ -valued $1$ -form $A$ on $M$ , the pullback connection on $N\times G$ corresponds to the pulled-back form $f^*A$ on $N$ .