Draft:Involutive Structures

• Comment: Reference 2 (Berhanu et al.) has an invalid ISBN and reference 5 (Eastwood et al.) has an invalid DOI. This makes verification untenable. (— 𝐬𝐝𝐒𝐝𝐬 — - talk) 12:43, 8 September 2025 (UTC)

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Involutive structures are a concept in differential geometry used to encode certain systems of first-order differential equations. They generalize the notion of the Cauchy–Riemann equations on complex manifolds and arise in the study of partial differential equations, complex geometry, and foliation theory.

Definition

Let $M$ be a smooth manifold. An involutive structure on $M$ is given by a complex subbundle ${\mathcal {D}}\subset TM\otimes \mathbb {C}$ of the complexified tangent bundle that is closed under the Lie bracket of vector fields. That is, if $X$ and $Y$ are smooth sections of ${\mathcal {D}}$ then the Lie bracket $[X,Y]$ is also a section of ${\mathcal {D}}$.

Integrability

Let ${\mathcal {D}}$ have (complex) rank $k$. The involutive structure ${\mathcal {D}}$ is called integrable if for every point $p\in M$ there exists a neighbourhood $U$ of $p$ and smooth functions $f_{1},\dots ,f_{k}$ on $U$ such that $df_{i}({\mathcal {D}})=0$ for each $i$ and $df_{1}\wedge \dots \wedge df_{k}\neq 0$ on $U$. In other words, locally there exist $k$ independent smooth first integrals whose differentials annihilate ${\mathcal {D}}$.

Examples

• A complex structure on a manifold of dimension $2n$ can be described as an involutive structure ${\mathcal {D}}$ of rank $n$ such that ${\mathcal {D}}\cap {\overline {\mathcal {D}}}=0$. In this case, one can write ${\mathcal {D}}$ as the $-i$-eigenspace of an almost complex structure $J$, and involutivity is equivalent to the vanishing of the Nijenhuis tensor of $J$. A smooth function $f$ is holomorphic if and only if $df({\mathcal {D}})=0$. The Newlander–Nirenberg theorem states that complex structures are always integrable.

• A real involutive structure is an involutive structure ${\mathcal {D}}$ with ${\mathcal {D}}={\overline {\mathcal {D}}}$. In this case, one can view ${\mathcal {D}}$ as a subbundle of the real tangent bundle. The classical Frobenius theorem states that real involutive structures are always integrable.

• In analytic continuation problems, involutive structures can be used to establish extension theorems. For example, Eastwood and Graham constructed an involutive structure on the blow-up of $\mathbb {R} ^{n}$ in $\mathbb {C} ^{n}$ and used it to prove edge-of-the-wedge type theorems.

See also

• Distribution (differential geometry)
• Almost complex structure
• Frobenius theorem (differential topology)
• CR structure
• Newlander–Nirenberg theorem