A smooth map between manifolds pulls a function back by composition and pulls a covariant tensor back by applying its differential to each vector argument. The pullback of a differential form is the alternating-tensor case and commutes with the exterior derivative.
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In differential geometry, a pullback is an important operation that allows you to relate the geometry of different manifolds by transferring differential forms, functions, or vector fields from one manifold to another through a smooth map. Given two smooth manifolds \( M \) and \( N \), and a smooth map \( f: N \to M \), the pullback operation can be applied in various contexts, most commonly with differential forms.