How Differential Forms Restrict to Submanifolds

A differential form defined on a manifold $M$ can be used on a smooth submanifold $S\subseteq M$, but the correct operation is not usually a pointwise set-theoretic restriction. The standard construction is the pullback along the inclusion of $S$ into $M$.

If

$$i:S\hookrightarrow M$$

is the inclusion map and $\omega$ is a differential $k$-form on $M$, then the induced form on $S$ is

$$i^*\omega.$$

This is a $k$-form on $S$. It takes $k$ tangent vectors to $S$, regards them as tangent vectors to $M$, and evaluates $\omega$ on them.

The definition through tangent vectors

At a point $p\in S$, the tangent space $T_pS$ is naturally a subspace of $T_pM$. For $v_1,\ldots,v_k\in T_pS$, define

$$(i^*\omega)_p(v_1,\ldots,v_k)=\omega_p(v_1,\ldots,v_k).$$

The right-hand side makes sense because each vector tangent to $S$ is also tangent to $M$.

More generally, if $F:S\to M$ is any smooth map, not necessarily an inclusion, then

$$(F^*\omega)_p(v_1,\ldots,v_k) =\omega_{F(p)}\bigl(dF_p(v_1),\ldots,dF_p(v_k)\bigr).$$

Restriction to a submanifold is the special case $F=i$.

For a smooth function $f$ on $M$, this gives

$$i^*f=f|_S,$$

the ordinary restriction of $f$. For one-forms, however, the derivative directions must also be restricted to tangent directions of $S$.

Coordinate formula

Suppose $S$ is parametrized by a smooth map

$$\phi:U\subseteq\mathbb{R}^r\to M,$$

whose image lies in $S$. To calculate the form on $S$, pull it back by $\phi$.

For example, if $\omega$ is a one-form in coordinates $(x^1,\ldots,x^n)$,

$$\omega=\sum_{j=1}^n a_j(x)\,dx^j,$$

and the parametrization is

$$\phi(u^1,\ldots,u^r)=\bigl(x^1(u),\ldots,x^n(u)\bigr),$$

then

$$\phi^*\omega =\sum_{j=1}^n a_j(\phi(u))\,d(x^j\circ\phi) =\sum_{j=1}^n a_j(\phi(u))\sum_{\ell=1}^r \frac{\partial x^j}{\partial u^\ell}\,du^\ell.$$

Thus the ambient differentials $dx^j$ are replaced by their expressions along the parametrization.

For wedge products, pullback preserves multiplication:

$$\phi^*(\alpha\wedge\beta)=\phi^*\alpha\wedge\phi^*\beta.$$

It also commutes with exterior differentiation:

$$d(\phi^*\omega)=\phi^*(d\omega).$$

These identities are especially useful when working with differential forms on surfaces and curves.

Worked example: a plane in three-dimensional space

Let $S\subseteq\mathbb{R}^3$ be the plane $z=0$. Its inclusion can be written as

$$\phi(u,v)=(u,v,0).$$

Consider the ambient one-form

$$\omega=x\,dy+z\,dx+y\,dz.$$

Using $x=u$, $y=v$, $z=0$, and

$$dx=du,\qquad dy=dv,\qquad dz=0$$

on the plane, we obtain

$$\phi^*\omega=u\,dv.$$

Therefore the form induced on $S$ is $u\,dv$, after identifying the plane with the $(u,v)$-plane.

Now consider the two-form

$$\eta=dx\wedge dy+dy\wedge dz.$$

Its restriction is

$$\phi^*\eta=du\wedge dv+dv\wedge 0=du\wedge dv.$$

The term $dy\wedge dz$ disappears because the $z$-coordinate is constant along the plane.

A three-form on $\mathbb{R}^3$, such as

$$\tau=dx\wedge dy\wedge dz,$$

restricts to zero on the two-dimensional plane. There are only two independent tangent directions available, so three tangent vectors in $T_pS$ must be linearly dependent. Consequently,

$$\phi^*\tau=du\wedge dv\wedge 0=0.$$

Worked example: the sphere

Let $S^2\subseteq\mathbb{R}^3$ be the unit sphere, and consider the one-form

$$\omega=x\,dy-y\,dx.$$

Using spherical coordinates away from the poles,

$$\phi(\theta,\varphi) =(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta),$$

we have

$$\phi^*\omega=x\,d y-y\,d x.$$

The coordinate differentials are

$$dx=\cos\theta\cos\varphi\,d\theta-\sin\theta\sin\varphi\,d\varphi,$$ $$dy=\cos\theta\sin\varphi\,d\theta+\sin\theta\cos\varphi\,d\varphi.$$

Substitution gives

$$\begin{aligned} \phi^*\omega &=(\sin\theta\cos\varphi)\,dy -(\sin\theta\sin\varphi)\,dx\\ &=\sin^2\theta\,d\varphi. \end{aligned}$$

Thus the ambient form $x\,dy-y\,dx$ becomes $\sin^2\theta\,d\varphi$ on the sphere.

The same method works for a surface given implicitly by an equation such as $g(x,y,z)=0$. In that case, one may either choose a local parametrization or evaluate the ambient form directly on tangent vectors satisfying

$$dg_p(v)=0.$$

Indeed, the tangent space to a regular level surface is

$$T_pS=\ker(dg_p).$$

Differential forms that vanish on a submanifold

A form $\omega$ restricts to zero on $S$ precisely when

$$\omega_p(v_1,\ldots,v_k)=0$$

for every $p\in S$ and every $v_1,\ldots,v_k\in T_pS$.

For a one-form, this means that $\omega_p$ annihilates all tangent vectors to $S$. If $S$ is the level set $g=0$ of a regular function, then every one-form of the form

$$\omega=h\,dg$$

restricts to zero on $S$, because $dg(v)=0$ for every $v\in TS$.

For instance, on the plane $z=0$, the form $dz$ restricts to zero. Any form containing a factor of $dz$ also restricts to zero, such as $dx\wedge dz$.

Common mistakes

Confusing restriction with substituting coordinates only

For a one-form, replacing $z$ by $0$ in its coefficient functions is not enough. You must also replace the differentials according to the parametrization. On $z=0$, $dz$ becomes zero, while $dx$ and $dy$ remain tangent differentials.

Treating ambient vectors as arbitrary

The restricted form is evaluated only on vectors in $T_pS$. A vector normal to the submanifold is not an allowed input when evaluating a form on $S$.

Forgetting the degree bound

If $\dim S=r$, every $k$-form on $S$ with $k>r$ is zero. This follows because an alternating $k$-linear map on an $r$-dimensional vector space vanishes when $k>r$.

Assuming every ambient form is uniquely determined by its restriction

Different ambient forms can have the same restriction. On the plane $z=0$, both $dx$ and $dx+z\,dy+dz$ restrict to $dx$, since $z=0$ and $dz=0$ along tangent directions. Restriction loses information in normal directions.

Integration over submanifolds

Integration is defined using the same pullback. If $S$ is an oriented $r$-dimensional submanifold and $\omega$ is an $r$-form on $M$, then in a parametrized region $\phi:U\to S$,

$$\int_S\omega=\int_U\phi^*\omega,$$

with the appropriate orientation and a sum over parametrization patches when necessary. This is why pullback is the central operation for integrating ambient differential forms over curves, surfaces, and higher-dimensional submanifolds.