Fundamental vector fields for a left action

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $G$ act smoothly on the left of a smooth manifold $M$, let $\mathfrak{g}=T_{e}G$, and let $X\in \mathfrak{g}$. The fundamental vector field associated with $X$ is

$X_M(x):={\frac{d}{dt}\mid }_{t=0}{\exp }_G(-tX)\cdot x\in T_{x}M\mathrm{.}$

The minus sign is part of the standing convention. With it, the assignment $X\mapsto X_M$ is a Lie-algebra homomorphism for a left action; without it, the usual left-action infinitesimal generator is an antihomomorphism. The following theorem proves the bracket claim rather than building it into this definition.

The exponential map is smooth by The Lie-group exponential map is smooth with identity differential at zero, so $(t,x)\mapsto {\exp }_G(-tX)\cdot x$ is smooth. In local coordinates, differentiating this smooth map in the $t$-variable at $0$ gives coefficients that depend smoothly on $x$. Thus $x\mapsto X_M(x)$ is a smooth tangent-bundle section in the sense of A smooth vector field is a smooth section of the tangent bundle.

Here ${\mathrm{A}\mathrm{C}}_{\omega }$ is countable choice and is used through both the supplied exponential-map construction and the canonical smooth tangent-bundle structure underlying the smooth-section interface. For $X=0$ the field is zero. The definition applies to disconnected $G$ and $M$, and makes no effectiveness, freeness, or properness assumption on the action.