{title2=$l_\xi$}
{wiki=Invariant_vector_field}

For $\xi$ in the <Lie algebra> of $G$, the left-invariant vector field is $l_\xi(g)=(dL_g)_e\xi$. Its global <local flow>[flow] is right translation $\Phi^t=R_{\exp(t\xi)}$.

Left-invariant vector field

A <local flow> of $v$ is a smooth family $\Phi^t$ defined near $\{0\}\times X$ such that $\Phi^0=\operatorname{id}$, $\partial_t\Phi^t(x)=v(\Phi^t(x))$, and $\Phi^{s+t}=\Phi^s\circ\Phi^t$ whenever defined. The <Lie derivative of a differential form> is
$$
\mathcal L_v\alpha=\left.\frac d{dt}\right|_{t=0}(\Phi^t)^*\alpha.
$$
<Cartan's magic formula> is
$$
\boxed{\mathcal L_v\alpha=d(\iota_v\alpha)+\iota_v(d\alpha).}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 115 / 3 / a / Solution

{title2=$\Phi^t$}
{wiki=Flow_(mathematics)}

A local flow of a smooth <vector field> $V$ is a smooth map $\Phi:D\to M$, defined on an open neighborhood of $\{0\}\times M$, such that $\Phi^0(x)=x$, $\partial_t\Phi^t(x)=V(\Phi^t(x))$, and $\Phi^{s+t}=\Phi^s\circ\Phi^t$ whenever both sides are defined.

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 Incoming links: Local flow