Closed differential form

Let $M$ be a smooth manifold and let $\omega$ be a differential \\(k\\)-form on $M$. The notion of “closedness” is defined using the exterior derivative .

Definition

A form $\omega\in\Omega^k(M)$ is closed if

The vector space of closed $k$-forms is

Closed forms are the “cocycles” in the complex $(\Omega^\ast(M),d)$; passing to cohomology by quotienting out exact forms yields the de Rham cohomology groups .

Basic facts

• Every exact form is closed: if $\omega=d\eta$, then $d\omega=d(d\eta)=0$ because $d^2=0$.
• Closedness is preserved by pullback: if $F:M\to N$ is a smooth map and $\omega$ is closed on $N$, then the pullback $F^*\omega$ is closed on $M$.

Examples

1. Constant 1-forms on $\mathbb{R}^n$.
   In standard coordinates, each $dx^i$ is closed because $d(dx^i)=0$. Any constant-coefficient 1-form $\sum_i a_i\,dx^i$ is also closed.

2. Standard symplectic form on $\mathbb{R}^{2n}$.
   With coordinates $(x_1,y_1,\dots,x_n,y_n)$, the 2-form

   $$\omega_0=\sum_{i=1}^n dx_i\wedge dy_i$$

   is closed since $d(dx_i)=d(dy_i)=0$ and $d$ satisfies the graded Leibniz rule.

3. A closed but not exact 1-form on $\mathbb{R}^2\setminus\{0\}$.
   On $U=\mathbb{R}^2\setminus\{0\}$ with coordinates $(x,y)$, the 1-form

   $$\omega=\frac{-y\,dx + x\,dy}{x^2+y^2}$$

   is closed (it is the “angular” form). It is not exact on $U$, which can be detected by integrating $\omega$ around the unit circle: the integral is nonzero, so $\omega$ cannot be $d\eta$ globally on $U$.