Pullback (differential geometry)

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.