Lie derivative

Remarks

For a smooth function $f$, $\mathcal{L}_X f = X f$, the derivative of $f$ in the direction of $X$. For a vector field $Y$, $\mathcal{L}_X Y = [X, Y]$, the Lie bracket. For a differential form $\omega$, Cartan's formula gives $\mathcal{L}_X \omega = d(\iota_X \omega) + \iota_X d\omega$. Example: on $\R^2$ let $X = -y \, \partial_x + x \, \partial_y$ and $f = x^2 + y^2$; then $\mathcal{L}_X f = -y \cdot 2x + x \cdot 2y = 0$, as $f$ is invariant under the rotations $\varphi_t$. Non-example of tensoriality: $\mathcal{L}_X Y$ is not $C^\infty(M)$-linear in $X$, since $\mathcal{L}_{fX} Y = f [X, Y] - (Y f) X$. So $(\mathcal{L}_X T)_p$ depends on $X$ near $p$, not only on $X_p$; this is the difference with a covariant derivative. The Lie derivative is a derivation: $\mathcal{L}_X (S \otimes T) = \mathcal{L}_X S \otimes T + S \otimes \mathcal{L}_X T$, and it commutes with contractions.