Bundle isomorphism - Knowlpedia

Let $\pi:E\to M$ and $\pi':E'\to M'$ be smooth fiber bundles. A bundle isomorphism is a bundle morphism $(\Phi,f)$ where:

$f:M\to M'$ is a diffeomorphism,
$\Phi:E\to E'$ is a diffeomorphism,
and the projections commute: $\pi'\circ \Phi=f\circ \pi$ .

In local trivializations, a bundle isomorphism is locally a diffeomorphism of products that preserves the base coordinate: over $U\subset M$ , it is represented as $(x,u)\mapsto (f(x),\psi(x,u))$ with each $\psi(x,\cdot)$ a diffeomorphism of the typical fiber.

Equivalent characterizations

Equivalently, $\Phi$ is a diffeomorphism that is fiber-preserving over $f$ , and its inverse $\Phi^{-1}$ is fiber-preserving over $f^{-1}$ . If $M=M'$ and $f=\mathrm{id}_M$ , one often calls $\Phi$ a bundle automorphism.

Examples

Trivialization as an isomorphism: a global trivialization $E\cong M\times F$ is a bundle isomorphism from $E$ to the product bundle.
Tangent bundles: if $f:M\to N$ is a diffeomorphism, then $df:TM\to TN$ is a bundle isomorphism covering $f$ .
Gauge transformations on products: for $M\times F\to M$ , any map $(x,u)\mapsto(x,h(x)(u))$ with $h:M\to \mathrm{Diff}(F)$ defines a bundle automorphism.