Lie derivative - Knowlpedia

Let $M$ be a smooth manifold and let $X$ be a vector field on $M$ with local flow $\Phi_t$ , defined near $t=0$ by $\partial_t\Phi_t(p)=X_{\Phi_t(p)}$ and $\Phi_0(p)=p$ . For a differential k-form $\omega\in\Omega^k(M)$ , the Lie derivative of $\omega$ along $X$ is

\mathcal{L}_X\omega = \left.\frac{d}{dt}\right|_{t=0}\Phi_t^*\omega,

where $\Phi_t^*$ denotes pullback of forms by the diffeomorphism $\Phi_t$ .

Cartan formula

The Lie derivative is characterized by Cartan's formula:

\mathcal{L}_X\omega = d(\iota_X\omega) + \iota_X(d\omega),

where $\iota_X$ is the interior product and $d$ is the exterior derivative.

Examples

Functions. For $f\in C^\infty(M)$ , we have $\mathcal{L}_X f = X(f)$ , the directional derivative.

Rotation on the plane. On $\mathbb{R}^2$ , let $X = -y\partial_x + x\partial_y$ . For $\omega = dx\wedge dy$ , we have $\mathcal{L}_X\omega = 0$ .

Left-invariant forms. On a Lie group $G$ , if $\omega$ and $X$ are both left-invariant, then $\mathcal{L}_X\omega$ is left-invariant.