Editing Pullback (differential geometry) (section)

==Pullback of smooth functions and smooth maps==

Let <math>\phi:M\to N</math> be a smooth map between (smooth) manifolds <math>M</math> and <math>N</math>, and suppose <math>f:N\to\R</math> is a smooth function on <math>N</math>. Then the '''pullback''' of <math>f</math> by <math>\phi</math> is the smooth function <math>\phi^*f</math> on <math>M</math> defined by <math>(\phi^*f)(x)=f(\phi(x))</math>. Similarly, if <math>f</math> is a smooth function on an [[open set]] <math>U</math> in <math>N</math>, then the same formula defines a smooth function on the open set  <math>\phi^{-1}(U)</math>. (In the language of [[sheaf (mathematics)|sheaves]], pullback defines a morphism from the [[sheaf of smooth functions]] on <math>N</math> to the [[direct image sheaf|direct image]] by <math>\phi</math> of the sheaf of smooth functions on <math>M</math>.)

More generally, if <math>f:N\to A</math> is a smooth map from <math>N</math> to any other manifold <math>A</math>, then <math>(\phi^*f)(x)=f(\phi(x))</math> is a smooth map from <math>M</math> to <math>A</math>.