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 may consider an ideal of this collection; moreover, since \(\Omega^\ast(M)\) 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 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 \(\mathcal{D}\).

We then define the collection of differential forms that annihilate \(\mathcal{D}\) by

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

Proposition 2 Let \(\mathcal{D}\) be a \(k\)-dimensional distribution 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 immediate 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 \(p\). Adjoin vector fields \(X_1,\ldots, X_{m-k}\) to obtain a local basis \(\{X_1,\ldots, X_m\}\) for the tangent spaces on a neighborhood \(U\) of \(p\). Taking the dual basis 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 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, 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 \(\mathcal{D}\) 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 \(\Phi:N\rightarrow M\), we say that \(N\) is an integral manifold of an ideal \(\mathcal{I}\) if \((d\Phi)^\ast(\omega)\equiv 0\) for every \(\omega\in\mathcal{I}\).

Under these conditions, Frobenius’s theorem can be stated 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. Assume also that the collection of 1-forms 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 for any \(q_0\in N\) and \(p_0\in M\), there exist a suitable open neighborhood \(U\) of \(q_0\) and a \(C^\infty\) function \(f:U\rightarrow M\) with \(f(q_0)=p_0\) such that

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

   holds.

Proof

We briefly sketch the argument.

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

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

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

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

   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 \((d\iota)^\ast(\mu_i)=0\). 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 immediate.

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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