# Definition Below, we define pull-backs and push-forwards of various mathematical objects. 1. *Pull-back of a function $f$ by $\phi$* $\phi^* f = f \circ \phi$ 2. *Push-forward of a function $f$ by $\phi$* $\phi_* f = f \circ \phi^{-1}$ 3. *Push-forward of a vector field $X$ by $\phi$* $ \begin{align} (\phi_* X)(\phi(z)) &= \pmb{D}\phi(z) \cdot X(z),\\ (\phi_* X)^I &= \frac{\partial \phi^I}{\partial z^J} X^J \end{align} $ 4. *Push-forward of a vector field $X$ by $\phi$* $ \phi^* X = (\phi^{-1})_* X $ 5. *Pull-back of a bilinear form $\Omega$ on $Z$ by a function $\phi$* $ \begin{align} \left(\varphi^{*} \Omega\right)_{z}\left(z_{1}, z_{2}\right)&=\Omega\left(\mathbf{D} \varphi(z) \cdot z_{1}, \mathbf{D} \varphi(z) \cdot z_{2}\right)\\ \left(\varphi^{*} \Omega\right)_{I J}&=\frac{\partial \varphi^{K}}{\partial z^{I}} \frac{\partial \varphi^{L}}{\partial z^{J}} \Omega_{K L} \end{align} $ 6. *Push-forward of a bilinear form $\Omega$ on $Z$ by $\phi$ $ \phi_*\Omega = (\phi^{-1})^*\Omega $ # Example Let's work through an example from Nakahara. Let $V, W$ be [[Vector Space|vector spaces]], over a [[field]] $\mathbb{F}$, with a [[Linear Map|linear map]] $f:V \rightarrow W$ and let $g: W \rightarrow \mathbb{F}$ be a [[one-form|1-form]] on $W$ (i.e., $g \in W^*$, the [[Dual Space]]). As a result, we have the induced [[Composite Map]] $g \circ f: V \rightarrow \mathbb{F}$ such that $(g \circ f) \in V^*$ We also have an induced map $f^*: W^* \rightarrow V^*$ defined by $f^*:g \mapsto h \equiv (g \circ f) = f^*(g)$. The map $h \equiv (g \circ f)$ is known as the *pullback* of $g$ by $f^*$. ![[Pasted image 20210630115904.png]] In the diagram, $K = \mathbb{F}$.