Suppose $G$ is a Lie group and act smoothly on a compact oriented manifold $M$, $X\in\mathfrak{g}$ is an element in Lie algebra of $G$ such that the vector field $\tilde X$ generated by $X$ has isolated zeros, i.e.,

$ \tilde X(p)=\frac{d}{dt}\big\vert_{t=0}exp(tX)\cdot p $

Then $\tilde X$ act on space of vector field on $M$ by Lie bracket: $F\mapsto [\tilde X,F]$. And this action induces a well-defined action on $T_pM$ for any point $p$ with $\tilde X(p)=0$. This action is denoted as

$ L(X,p):T_pM\to T_pM $

$ ~~~~~~~~~~~~~~~~~~~~~~v\mapsto [\tilde X,F_v](p)~~~~\text{where } F_v\text{ is a vector field with }F_v(p)=v $

The question is to show that $L(X,p)$ is invertible.

I tend to show that if there is a non-zero vector $\xi\in T_pM$ s.t. $L(X,p)(\xi)=0$ then $exp(tX)$ fixes $exp_p(sv)$ for $t,s$ small. And this contradicts the fact that $p$ is isolated. But I don't know how to show this.


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