The Pointwise Pullback
A smooth map yields a linear map on tangent vectors, its differential, which pushes vectors
forward in the direction of the map. Dualizing this linear map produces an operation on
covectors running in the opposite direction. This is the manifold instance of the dual map
from linear algebra, and it is the operation that, unlike the pushforward of vectors, will
extend without obstruction to entire fields.
Let \(F : M \to N\) be a
smooth map
between smooth manifolds with or without boundary, and let \(p \in M\). Its
differential
\(dF_p : T_pM \to T_{F(p)}N\) is a linear map between tangent spaces. Applying the dual-map
construction to it produces a linear map between the cotangent spaces, in the reverse direction.
Definition: The Pointwise Pullback
Let \(F : M \to N\) be a smooth map and \(p \in M\). The (pointwise) pullback by \(F\) at
\(p\), also called the cotangent map of \(F\), is the dual of the
differential \(dF_p\),
\[
dF^*_p : T^*_{F(p)}N \longrightarrow T^*_pM,
\]
characterized by
\[
\big( dF^*_p(\xi) \big)(v) = \xi\big( dF_p(v) \big),
\quad \xi \in T^*_{F(p)}N, \quad v \in T_pM .
\]
The pointwise pullback is the
dual map
of \(dF_p\), specialized to the differential of a smooth map. It inherits the contravariance
recorded there. For a composition \(G \circ F\) of smooth maps, the chain rule for
differentials \(d(G \circ F)_p = dG_{F(p)} \circ dF_p\) dualizes to
\[
d(G \circ F)^*_p = dF^*_p \circ dG^*_{F(p)},
\]
and the differential of an identity map is an identity, so its pullback is too. The
assignment sending a pointed manifold to its cotangent space and a smooth map to its
cotangent map is thus a contravariant operation. It is the second concrete instance of the
dualization construction, here on pointed manifolds and their smooth maps. The first was the
dual of a linear map on finite-dimensional vector spaces. The two live on different
categories but share the same reversal of arrows.
Pullbacks of Covector Fields
The pointwise pullback assembles into an operation on entire covector fields, and here the
contrast with vector fields is sharp. A
pushforward of a vector field
is defined only in special cases: when \(F\) is a diffeomorphism, and for
left-invariant fields when \(F\) is a Lie group homomorphism, through the
induced Lie algebra homomorphism.
The obstruction is that two
points of \(M\) with the same image would in general be assigned conflicting vectors at
that image, while points of \(N\) outside the image would be assigned none. Covector fields suffer no such obstruction. They
always pull back, because the value of the pulled-back field at \(p\) is determined entirely
by the single covector at \(F(p)\), with no competition between preimages and no need to
cover the image.
Definition: Pullback of a Covector Field
Let \(F : M \to N\) be a smooth map between smooth manifolds with or without boundary, and let
\(\omega\) be a
rough covector field
on \(N\). The pullback of \(\omega\) by \(F\)
is the rough covector field \(F^*\omega\) on \(M\) whose value at each \(p \in M\) is the
pointwise pullback of \(\omega_{F(p)}\),
\[
(F^*\omega)_p = dF^*_p\big( \omega_{F(p)} \big).
\]
It acts on a tangent vector \(v \in T_pM\) by
\[
(F^*\omega)_p(v) = \omega_{F(p)}\big( dF_p(v) \big).
\]
The pullback interacts with multiplication by functions and with differentials in the
simplest possible way. Scalar factors pull back by composition, and the pullback commutes
with the differential.
Proposition: Pullback of Products and Differentials
Let \(F : M \to N\) be a smooth map between smooth manifolds with or without boundary. Suppose
\(u\) is a continuous real-valued function on \(N\) and \(\omega\) is a covector field on \(N\).
Then
\[
F^*(u\omega) = (u \circ F)\, F^*\omega.
\]
If in addition \(u\) is smooth, then
\[
F^*(du) = d(u \circ F) .
\]
Proof:
For the first identity, evaluate at \(p \in M\). The covector field \(u\omega\) has value
\((u\omega)_{F(p)} = u(F(p))\, \omega_{F(p)}\), where the scalar multiplies the
covector field
pointwise. Pulling back and using linearity of the pointwise pullback \(dF^*_p\),
\[
\begin{align*}
\big( F^*(u\omega) \big)_p
&= dF^*_p\big( (u\omega)_{F(p)} \big) \\\\
&= dF^*_p\big( u(F(p))\, \omega_{F(p)} \big) \\\\
&= u(F(p))\, dF^*_p\big( \omega_{F(p)} \big) \\\\
&= (u \circ F)(p)\, (F^*\omega)_p,
\end{align*}
\]
which is the value at \(p\) of \((u \circ F)\, F^*\omega\).
For the second identity, let \(v \in T_pM\) be arbitrary and unwind the definitions:
\[
\begin{align*}
(F^*du)_p(v)
&= \big( dF^*_p(du_{F(p)}) \big)(v) \\\\
&= du_{F(p)}\big( dF_p(v) \big) \\\\
&= dF_p(v)\, u \\\\
&= v(u \circ F) \\\\
&= d(u \circ F)_p(v).
\end{align*}
\]
Here the third equality is the definition of the differential \(du\). The fourth is the defining
property of \(dF_p\), applied to \(v\) and \(u\), and the last is the definition of
\(d(u \circ F)\). Since \(v\) was arbitrary, \(F^*du = d(u \circ F)\).
Smoothness of the pulled-back field follows once its coordinate expression is in hand, which the next
section provides.
Proposition: Smoothness of the Pullback
Let \(F : M \to N\) be a smooth map between smooth manifolds with or without boundary, and let
\(\omega\) be a covector field on \(N\). Then \(F^*\omega\) is a continuous covector field on
\(M\), and it is smooth whenever \(\omega\) is smooth.
The Coordinate Formula
The two properties of the pullback combine into a formula that makes computation in coordinates
immediate. A covector field written in coordinates lives only on the coordinate domain
\(V \subseteq N\), so the pullback below is understood on \(F^{-1}(V)\), taken along the
restriction of \(F\) to \(F^{-1}(V)\).
Proof:
Apply the product and differential rules for the pullback to \(\omega = \omega_j\, dy^j\):
\[
\begin{align*}
F^*\omega &= F^*\big( \omega_j\, dy^j \big) \\\\
&= (\omega_j \circ F)\, F^*\big( dy^j \big) \\\\
&= (\omega_j \circ F)\, d\big( y^j \circ F \big),
\end{align*}
\]
using the pointwise linearity of \(dF^*_p\) to split the sum over \(j\), then
\(F^*(u\omega) = (u \circ F)\, F^*\omega\) for the second equality and
\(F^*(du) = d(u \circ F)\) with \(u = y^j\) for the third. Identifying
\(y^j \circ F = F^j\) gives the stated formula.
No general transition formula need be memorized. The substitution stated above carries all
the information.
This expression also settles the regularity left open above. The component functions
\(\omega_j \circ F\) are continuous, and smooth when \(\omega\) is smooth, since \(F\) is
smooth. The differentials \(dF^j = d(y^j \circ F)\) are smooth covector fields, being
differentials of smooth functions. A sum of products of such fields is continuous in general
and smooth when the coefficients are, so \(F^*\omega\) is a continuous covector field,
smooth whenever \(\omega\) is smooth. Coordinate domains cover \(N\), and continuity and smoothness are local
conditions, so this holds on all of \(M\) and completes the proof of the smoothness
proposition stated previously.
Pullback by substitution: two computations
A map between Euclidean spaces. Let \(F : \mathbb{R}^3 \to \mathbb{R}^2\) be
\((u, w) = F(x, y, z) = (x^2 y,\, y \sin z)\), and let \(\omega = u\, dw + w\, du\) on
\(\mathbb{R}^2\). Substituting \(u = x^2 y\) and \(w = y \sin z\) everywhere and expanding by the
rules above,
\[
\begin{align*}
F^*\omega &= (u \circ F)\, d(w \circ F) + (w \circ F)\, d(u \circ F) \\\\
&= x^2 y \, d(y \sin z) + y \sin z \, d\big( x^2 y \big) \\\\
&= 2xy^2 \sin z \, dx + 2x^2 y \sin z \, dy + x^2 y^2 \cos z \, dz.
\end{align*}
\]
A change of coordinates. Let \((r, \theta)\) be polar coordinates on the right
half-plane, related to Cartesian coordinates by the diffeomorphism
\(P(r, \theta) = (r \cos\theta,\, r \sin\theta)\). Pulling the covector field \(x\, dy - y\, dx\)
back along \(P\) is once more a substitution, \(x = r\cos\theta\) and \(y = r\sin\theta\):
\[
\begin{align*}
P^*\big( x\, dy - y\, dx \big)
&= (r \cos\theta)\, d(r \sin\theta) - (r \sin\theta)\, d(r \cos\theta) \\\\
&= (r \cos\theta)(\sin\theta \, dr + r \cos\theta \, d\theta)
- (r \sin\theta)(\cos\theta \, dr - r \sin\theta \, d\theta) \\\\
&= r^2 \, d\theta.
\end{align*}
\]
Because \(P\) is precisely the map that re-expresses one chart in terms of the other, its
pullback is what is meant by writing the same covector field in the new coordinates. The
Euclidean pullback of the first computation and the coordinate change here are two
instances of a single operation.
Restriction to Submanifolds
The most frequent pullback is by an inclusion map. Whether a vector field restricts to a
submanifold is a delicate question, since a vector tangent to the ambient manifold need not
be tangent to the submanifold. For covector fields the situation is far simpler. A covector
on the ambient manifold can always be evaluated on vectors that happen to be tangent to the
submanifold, and recording only those values is exactly the pullback by the inclusion.
Let \(M\) be a smooth manifold with or without boundary, let \(S \subseteq M\) be an
immersed submanifold
with or without boundary, and let \(\iota : S \hookrightarrow M\) be the inclusion map. For
a smooth covector field \(\omega\) on \(M\), the pullback \(\iota^*\omega\) is a smooth
covector field on \(S\). Its meaning is read off the definition: for \(p \in S\) and
\(v \in T_pS\),
\[
(\iota^*\omega)_p(v) = \omega_p\big( d\iota_p(v) \big) = \omega_p(v) .
\]
The last equality uses the
realization of \(T_pS\) as a subspace of \(T_pM\).
Under it the differential \(d\iota_p\) of the inclusion is the injection of that subspace,
carrying a vector \(v \in T_pS\) to the same vector regarded as an element of \(T_pM\).
Definition: Restriction of a Covector Field to a Submanifold
With \(\iota : S \hookrightarrow M\) the inclusion of an immersed submanifold, the pullback
\(\iota^*\omega\) of a covector field \(\omega\) on \(M\) is called the restriction of
\(\omega\) to \(S\). It is the covector field on \(S\) whose value at each \(p \in S\)
is \(\omega_p\) evaluated only on vectors tangent to \(S\).
The word "restriction" must be handled with care, because the restricted field can vanish at
a point of \(S\) where the original covector field does not. The value \((\iota^*\omega)_p\)
discards all information about how \(\omega_p\) acts on vectors transverse to \(S\). If
\(\omega_p\) happens to be nonzero only on such transverse directions, its restriction is
zero even though \(\omega_p \neq 0\).
Restriction can vanish where the field does not
Take \(\omega = dy\) on \(\mathbb{R}^2\), and let \(S\) be the \(x\)-axis, an embedded
submanifold. As a covector field on \(\mathbb{R}^2\), \(\omega\) is nowhere zero, since one of its
component functions is the constant \(1\). Yet the coordinate function \(y\) is identically zero
on \(S\), so its restriction \(y \circ \iota\) is the zero function on \(S\), and therefore
\[
\iota^*\omega = \iota^*(dy) = d(y \circ \iota) = d(0) = 0 .
\]
The restriction vanishes identically while \(\omega\) itself vanishes nowhere. The reconciliation
is that \(dy\) measures the rate of change in a direction transverse to the \(x\)-axis, and that
direction is invisible to vectors tangent to \(S\).
This forces a distinction between two readings of the phrase "\(\omega\) vanishes on \(S\)". One says
that \(\omega\) vanishes along \(S\), or vanishes at points of \(S\), when
\(\omega_p = 0\) as a covector on the ambient tangent space \(T_pM\) for every \(p \in S\). The
weaker condition \(\iota^*\omega = 0\) is expressed by saying that the
restriction of \(\omega\) to \(S\) vanishes, or that the pullback of \(\omega\) to \(S\)
vanishes. The example shows the second can hold without the first. Vanishing along \(S\) is a
statement about \(\omega\) as a field on \(M\). Vanishing of the restriction is a statement
about its pullback to \(S\).
Restriction by inclusion is the mechanism that will later equip a submanifold with structure
inherited from its ambient manifold. When the ambient manifold carries a field that measures lengths
and angles of tangent vectors, pulling that field back along the inclusion records its action on the
vectors tangent to the submanifold, and so endows the submanifold with a measurement of its own. The
pullback developed here, applied to such a field, is the construction that produces the induced
geometry of a submanifold.