Bundle isomorphism

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: $\pi'\circ \Phi=f\circ \pi$.

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.

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.

Examples

1. Trivialization as an isomorphism: a global trivialization $E\cong M\times F$ is a bundle isomorphism from $E$ to the product bundle.
2. Tangent bundles: if $f:M\to N$ is a diffeomorphism, then $df:TM\to TN$ is a bundle isomorphism covering $f$.
3. 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.