Pullback of covectors

The contravariant map on cotangent spaces induced by a smooth map, defined by precomposing with the differential.

Pullback of covectors

Let $F:M\to N$ be a smooth map between smooth manifolds . For each $p\in M$ , the differential gives a linear map

dF_p:T_pM\to T_{F(p)}N

between tangent spaces .

Definition (pullback on a single fiber). The pullback of covectors at $p$ is the linear map

F_p^*:T_{F(p)}^*N \to T_p^*M

defined by

(F_p^*\alpha)(v):=\alpha\bigl(dF_p(v)\bigr) \quad\text{for }\alpha\in T_{F(p)}^*N,\; v\in T_pM.

Thus $F_p^*$ is the dual map of $dF_p$ (it is contravariant: it goes in the opposite direction).

This construction is fiberwise for the cotangent bundle : a covector at $F(p)$ pulls back to a covector at $p$ .

Functoriality. If $G:N\to P$ is another smooth map, then for each $p\in M$ ,

(G\circ F)_p^* = F_p^*\circ G_{F(p)}^*.

If $F$ is a diffeomorphism , then $dF_p$ is an isomorphism and $F_p^*$ is an isomorphism with inverse given by the corresponding pullback along $F^{-1}$ .

Pullback of covectors is the $k=1$ case of the pullback of differential forms .

Examples

Pulling back the standard covectors on $\mathbb{R}^2$ . Let $F:\mathbb{R}^2\to\mathbb{R}^2$ be given by $F(u,v)=(u^2,uv)$ , and let $(x,y)$ be coordinates on the target. Then, viewing $dx$ and $dy$ as smooth covector fields,
$F^*(dx)=d(u^2)=2u\,du,\qquad F^*(dy)=d(uv)=v\,du+u\,dv.$
At a specific point $(u,v)$ , this matches the fiberwise definition $\alpha\mapsto \alpha\circ dF_{(u,v)}$ .
Inclusion of the circle. Let $i:S^1\hookrightarrow\mathbb{R}^2$ be the inclusion and parametrize $S^1$ by $\gamma(\theta)=(\cos\theta,\sin\theta)$ . Then
$i^*(dx) = -\sin\theta\,d\theta,\qquad i^*(dy)=\cos\theta\,d\theta,$
since $x\circ\gamma=\cos\theta$ and $y\circ\gamma=\sin\theta$ .
Constant map. If $c:M\to N$ is constant with value $q\in N$ , then $dc_p=0$ for every $p$ , hence $c_p^*(\alpha)=0$ for every $\alpha\in T_q^*N$ .