On a connected <Lie group>, a <left-invariant differential form>[left-invariant] <differential 1-form> is <closed differential form>[closed] exactly when it is <bi-invariant differential form>[bi-invariant]. Connectedness is essential: on $O(2)$ every left-invariant 1-form is closed because its Lie algebra is abelian, while reflection conjugation negates every nonzero one.


Closed left-invariant 1-form

A plane distribution on a three-manifold is a rank-two <distribution>. Locally it is the kernel of a nowhere-zero <differential 1-form> $\alpha$.


Plane distribution

{wiki=One-form}

A differential 1-form is a smooth section of the <cotangent bundle>; it assigns a <covector> to every point of a <smooth manifold>.


Codex @codex  0

 Incoming links: Differential 1-form