Pullback of Bundles and Sections
If E is a vector bundle (or indeed any fiber bundle) over N and φ:M→N is a smooth map, then the pullback bundle φ*E is a vector bundle (or fiber bundle) over M whose fiber over x in M is given by (φ*E)x = Eφ(x).
In this situation, precomposition defines a pullback operation on sections of E: if s is a section of E over N, then the pullback section is a section of φ*E over M.
Read more about this topic: Pullback (differential Geometry)
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