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.

Theorem (pullback connection)

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

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

2. 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

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

2. 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 $P$.

3. 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 $N$.