Pullback of Smooth Functions and Smooth Maps
Let φ:M→ N be a smooth map between (smooth) manifolds M and N, and suppose f:N→R is a smooth function on N. Then the pullback of f by φ is the smooth function φ*f on M defined by (φ*f)(x) = f(φ(x)). Similarly, if f is a smooth function on an open set U in N, then the same formula defines a smooth function on the open set φ-1(U) in M. (In the language of sheaves, pullback defines a morphism from the sheaf of smooth functions on N to the direct image by φ of the sheaf of smooth functions on M.)
More generally, if f:N→A is a smooth map from N to any other manifold A, then φ*f(x)=f(φ(x)) is a smooth map from M to A.
Read more about this topic: Pullback (differential Geometry)
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