Definition of pull-back analogous to push-forward Let $U\subseteq\mathbb{R}^n, V\subseteq\mathbb{R}^m$ be open subsets, and $f:U\rightarrow V$ a $C^1$ map. Let $u,v$ be vector fields on $U,V$ respectively. Then the **push-forward** $f_*u$ is equal to $v$ if and only if $u$ and $v$ are $f$-related. The vector fields $u$ and $v$ are **$f$-related** if for all $p\in U$ and $q=f(p)$, we have $Df(p)\cdot u(p) = v(q)$. How do we define the **pull-back**? Is it defined as the opposite of the push-forward like: $f^*v$ is equal to $u$ if and only if $u$ and $v$ are $f$-related?

Since there is a natural Riemannian metric on $\mathbb R^n$ and $\mathbb R^m$, a notion of pull back for vector fields could be defined as follows: let $v$ be a vector field on $V$. Then $f^*v$ is a vector field on $U$ so that for all vector fields $w$ on $U$ we have $\langle f^* v,w\rangle = \langle v, f_* w \rangle$, where $\langle\cdot,\cdot\rangle$ is the inner product on either $\mathbb R^n$ or $\mathbb R^m$ as appropriate. But since we are regarding vector fields as being dual spaces to the space of vector fields, it is more common to call them cotangent vector fields, or 1-forms.