Differential Ideal

This post was translated from Korean by LLM (Kimi). The translation may contain errors or awkward sentences. The Korean original is the source of truth.

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

Definition of a Differential Ideal

We begin with the following definition.

Definition 1 Let \(\mathcal{D}\) be a \(k\)-dimensional distribution defined on a manifold \(M\). 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 each homogeneous part of \(\omega\) annihilates \(\mathcal{D}\).

In this case, we define the collection of differential forms that annihilate \(\mathcal{D}\) as

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

Proposition 2 Let \(\mathcal{D}\) be a \(k\)-dimensional distribution defined on a manifold \(M\).

1. \(\mathcal{I}(\mathcal{D})\) is an ideal of \(\Omega^\ast(M)\).
2. \(\mathcal{I}(\mathcal{D})\) is locally generated by \(m-k\) 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 obvious from the definition.

To prove the second claim, let \(p\in M\). Then there exist independent vector fields \(X_{m-k+1},\ldots, X_m\) generating \(\mathcal{D}\) in a neighborhood of \(p\). Now add vector fields \(X_1,\ldots, X_{m-k}\) to this set so that \(\{X_1,\ldots, X_m\}\) becomes a local basis for the tangent spaces in a neighborhood \(U\) of \(p\). Then, taking their duals, we obtain \(1\)-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 above arguments.

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

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

Differential Ideals and Frobenius’s Theorem

We now restate Frobenius’s theorem, which we examined in the previous post, in the language of differential forms. This is nothing more than converting the statement about vector fields into one about their dual \(1\)-forms.

Proposition 4 For a distribution \(\mathcal{D}\) defined on a manifold \(M\), 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 \(\Phi:N\rightarrow M\), \(N\) is called an integral manifold of an ideal \(\mathcal{I}\) if \((d\Phi)^\ast(\omega)\equiv 0\) holds for every \(\omega\in\mathcal{I}\).

Then under these conditions, Frobenius’s theorem can be written as follows.

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

Graphs and Differential Forms

The following theorem is an important tool for exploring the relationship between Lie groups and Lie algebras.

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

1. For any \(f:N\rightarrow M\), the graph \(\graph(f)\) is an integral manifold of the ideal \(\mathcal{I}\) generated by the set

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

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

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

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

   \[(df)^\ast(\omega_i)=\alpha_i|_U\]

   holds.

Proof

We briefly sketch the idea.

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

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

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

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

   given by

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

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

   satisfy \((d\iota)^\ast(\omega_i)=0\). This follows from

   \[(d\iota)^\ast(\mu_i)=(d(\pi_1\circ\iota))^\ast(df)^\ast(\omega_i)-(d(\pi_2\circ\iota))^\ast(\omega_i)=(df)^\ast(\omega_i)-(df)^\ast(\omega_i)=0\]

   which is clear.

2. Let \(\mathcal{I}\) be the ideal generated by the given set of forms. Then by Frobenius’s theorem, \(\mathcal{I}\) has an integral manifold \(I\) of dimension \((m+n)-m=c\). It then suffices to show that the restriction of \(d\pi_1\) to \(I_q\) is a bijection for an arbitrary point \(q\in I\).

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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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