Although when considering integration on manifolds we considers not being limited to a specific metric, differential manifolds have still not been singled out.
The situation with fiber bundles is similar; the codomain and its symmetries seem uncertain in the general case.
Example
homogeneous space
frame bundle
principal-bundle-connection
invariant δ isotropy-group & orbit decomposition at every point
This intuition is inspired by the specific symmetry space with the triple as a fiber bundle