Differential Ideal

We previously verified that the collection of differential forms constitutes a \(C^\infty(M)\)-algebra. Algebraically, one may consider an ideal of this collection; moreover, since this collection \(\Omega^\ast(M)\) is a dg-algebra, in the precise context it is more natural to consider the differential ideal that we define now.

Definition of a Differential Ideal

We begin with the following definition.

Definition 1 On a manifold \(M\), consider a \(k\)-dimensional distribution \(\mathcal{D}\). An \(l\)-form \(\omega\) is said to annihilate \(\mathcal{D}\) if for each \(p\in M\),

\[\omega_p(v_1,\ldots, v_l)=0,\qquad v_i\in\mathcal{D}(p)\]

holds. More generally, an arbitrary form \(\omega\) is said to annihilate \(\mathcal{D}\) if the homogeneous parts of \(\omega\) each annihilate \(\mathcal{D}\).

In this case, the collection of differential forms that annihilate \(\mathcal{D}\) is defined by

\[\mathcal{I}(\mathcal{D})=\{\omega\in\Omega^\ast(M)\mid\text{$\omega$ annihilates $\mathcal{D}$}\}\]

Proposition 2 On an \(m\)-dimensional manifold \(M\), consider a \(k\)-dimensional \(C^\infty\) distribution \(\mathcal{D}\).

1. \(\mathcal{I}(\mathcal{D})\) is an ideal of \(\Omega^\ast(M)\).
2. \(\mathcal{I}(\mathcal{D})\) is locally generated by \(m-k\) independent 1-forms.
3. If \(\mathcal{I}\) is an ideal satisfying the two conditions above, then there exists a unique \(k\)-dimensional distribution \(\mathcal{D}\) such that \(\mathcal{I}=\mathcal{I}(\mathcal{D})\).

Proof

The first claim is immediate from the definition.

To show the second claim, let \(p\in M\). Then in a neighborhood of \(p\), there exist independent vector fields generating \(\mathcal{D}\), \(X_{m-k+1},\ldots, X_m\). Now adjoin vector fields \(X_1,\ldots, X_{m-k}\) to this set, and suppose that \(\{X_1,\ldots, X_m\}\) forms a local basis for the tangent spaces on a neighborhood of \(p\), \(U\). Then taking their duals gives \(1\)-forms \(\omega_1,\ldots, \omega_m\), and since

\[\omega_i(X_j)=\delta_{ij}\]

it is easy to see that \(\omega_1,\ldots,\omega_{m-k}\) are the desired \(1\)-forms.

The third claim follows by reversing the arguments above.

Since \(\Omega^\ast(M)\) is a differential graded algebra, it is natural to focus on ideals that are also closed under the differential \(d:\Omega^\ast(M)\rightarrow\Omega^\ast(M)\).

Definition 3 In \(\Omega^\ast(M)\), an ideal \(\mathcal{I}\) is called a differential ideal if \(\mathcal{I}\) is closed under \(d\).

Differential Ideals and Frobenius’s Theorem

We now restate Frobenius’s theorem, examined in the previous post, in the language of differential forms. This is nothing special, but merely converting the statements about vector fields into those about their dual \(1\)-forms.

Proposition 4 On a manifold \(M\), for a \(C^\infty\) distribution \(\mathcal{D}\), the condition that \(\mathcal{D}\) is involutive is equivalent to \(\mathcal{I}(\mathcal{D})\) being a differential ideal.

Definition 5 For a manifold \(M\) and a submanifold \(\Phi:N\rightarrow M\), we say that \(N\) is an integral manifold of an ideal \(\mathcal{I}\) if for every \(\omega\in\mathcal{I}\), \((\dd{\Phi})^\ast(\omega)\equiv 0\) holds.

Under these conditions, Frobenius’s theorem can be stated as follows.

Theorem 6 Suppose that an \(m\)-dimensional manifold \(M\) and, generated by \(m-k\) independent 1-forms, a differential ideal \(\mathcal{I}\) are given. Then for each \(p\in M\), passing through \(p\) there exists an integral manifold of \(\mathcal{I}\), and this integral manifold is \(k\)-dimensional.

Graphs and Differential Forms

The following theorem is crucially used in exploring the relationship between Lie groups and Lie algebras.

Theorem 7 Let two manifolds \(M^m,N^n\) be given, and let \(\pi_1:N\times M\rightarrow N\) and \(\pi_2:N\times M\rightarrow M\) be the canonical projections, respectively. Assume also that the collection of 1-forms defined on \(M\) has a basis \(\{\omega_1,\ldots,\omega_m\}\).

1. For any \(C^\infty\) function \(f:N\rightarrow M\), \(\graph(f)\) is, with respect to the following set

   \[\{(\dd{(f\circ \pi_1)})^\ast(\omega_i)-(\dd{\pi_2})^\ast(\omega_i)\mid i=1,\ldots, m\}\]

   an integral manifold of the ideal \(\mathcal{I}\) generated by it.

2. For 1-forms \(\alpha_1,\ldots,\alpha_m\) on \(N\), assume that the ideal generated by the following set

   \[\{(\dd{\pi_1})^\ast(\alpha_i)-(\dd{\pi_2})^\ast(\omega_i)\mid i=1,\ldots,m\}\]

   is a differential ideal. Then whenever any \(q_0\in N,p_0\in M\) are given, there exist a suitable open neighborhood of \(q_0\), \(U\), and, satisfying \(f(q_0)=p_0\), a \(C^\infty\) function \(f:U\rightarrow M\) such that

   \[(\dd{f})^\ast(\omega_i)=\alpha_i\vert_U\]

   holds.

Proof

We only briefly sketch the flow of the argument.

1. First, following §Examples of Differentiable Manifolds, ⁋Example 5, we show that the following set

   \[\graph(f)=\{(p,q)\mid f(p)=q\}\]

   is a submanifold of \(N\times M\). Here, the inclusion map is given by the natural map

   \[\iota:\graph(f)\rightarrow N\times M;\qquad (p,q)\mapsto (p,q)\]

   Now, to prove that \(\mathcal{I}\) has this as an integral manifold, we must show that the given forms

   \[\mu_i:=(\dd{(f\circ\pi_1)})^\ast(\omega_i)-(\dd{\pi_2})^\ast(\omega_i)\]

   satisfy \((\dd{\iota})^\ast(\mu_i)=0\). This follows from

   \[(\dd{\iota})^\ast(\mu_i)=(\dd{(\pi_1\circ\iota)})^\ast(\dd{f})^\ast(\omega_i)-(\dd{(\pi_2\circ\iota)})^\ast(\omega_i)=(\dd{f})^\ast(\omega_i)-(\dd{f})^\ast(\omega_i)=0\]

   which is immediate.

2. Let \(\mathcal{I}\) be the ideal generated by the given set of forms. Then by Frobenius’s theorem, \(\mathcal{I}\) has an \((m+n)-m=n\)-dimensional integral manifold \(I\). Afterwards, for any point \(q\in I\), it suffices to show that the restriction of \(\dd{\pi_1}\) to \(T_qI\) is a bijection.

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References

[War] Frank W. Warner. Foundations of Differentiable Manifolds and Lie Groups, Graduate texts in mathematics, Springer, 2013
[Lee] John M. Lee. Introduction to Smooth Manifolds, Graduate texts in mathematics, Springer, 2012

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