Formal Definition
Let π : E → B be a fiber bundle with abstract fiber F and let f : B′ → B be a continuous map. Define the pullback bundle by
and equip it with the subspace topology and the projection map π′ : f*E → B′ given by the projection onto the first factor, i.e.,
The projection onto the second factor gives a map such that the following diagram commutes:
If (U, φ) is a local trivialization of E then (f−1U, ψ) is a local trivialization of f*E where
It then follows that f*E is a fiber bundle over B′ with fiber F. The bundle f*E is called the pullback of E by f or the bundle induced by f. The map is then a bundle morphism covering f.
Read more about this topic: Pullback Bundle
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