Closed differential form

Let $M$ be a smooth manifold. A differential $k$ -form $\omega\in\Omega^k(M)$ is closed if its exterior derivative vanishes:

d\omega = 0.

The vector space of closed $k$ -forms is

Z^k(M) \coloneqq \ker\!\bigl(d:\Omega^k(M)\to\Omega^{k+1}(M)\bigr).

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

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.

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.

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 $U$ .