Construction: pullback principal bundle

Let $\pi:P\to M$ be a principal G-bundle with right action, and let $f:N\to M$ be a smooth map between smooth manifolds .

Construction (Pullback bundle)

Define the fiber product

Let $\pi_N:f^*P\to N$ be projection to the first factor, $\pi_N(y,p)=y$. Define a right $G$-action on $f^*P$ by

Then:

1. $f^*P$ is a smooth manifold and $\pi_N:f^*P\to N$ is a smooth submersion.
2. $(f^*P,\pi_N)$ is a principal $G$-bundle over $N$.
3. There is a canonical $G$-equivariant map $\widetilde f:f^*P\to P$ given by $\widetilde f(y,p)=p$ covering $f$ (i.e. $\pi\circ \widetilde f = f\circ \pi_N$).

This construction is functorial: if $g:K\to N$ is another smooth map, then $(f\circ g)^*P$ is canonically isomorphic to $g^*(f^*P)$ as principal bundles.

Examples

1. Restriction to a submanifold. If $i:U\hookrightarrow M$ is the inclusion of an open set (or an embedded submanifold), then $i^*P\to U$ is the restriction of $P$ to $U$.

2. Pullback of a trivial bundle. If $P\cong M\times G$, then $f^*P\cong N\times G$ canonically.

3. Pullback of local trivializations. If $P$ is described by transition functions $g_{ij}$ on a cover of $M$, then $f^*P$ is described by the pulled-back transition functions $g_{ij}\circ f$ on the induced cover $\{f^{-1}(U_i)\}$ of $N$.