Pullback কী?

Seja $\varphi\colon U \to V$ suave ($U \subseteq \R^m$, $V \subseteq \R^n$ abertos). O pullback $\varphi^*\colon \Omega^k(V) \to \Omega^k(U)$ é definido ponto a ponto pelo pullback linear ao longo da diferencial: $(\varphi^*\omega)_x = (D\varphi(x))^*\,\omega_{\varphi(x)}$. Concretamente, $\varphi^*$ substitui: $\varphi^*f = f \circ \varphi$ em funções, $\varphi^*(\dd y_i) = \dd\varphi_i = \sum_j\frac{\partial\varphi_i}{\partial x_j}\dd x_j$, e $\varphi^*(a\,\dd y_{i_1}\wedge\dots\wedge \dd y_{i_k}) = (a\circ\varphi)\,\dd\varphi_{i_1}\wedge\dots\wedge \dd\varphi_{i_k}$.