Suppose a manifold $M$ admits a smooth Lie Group action $G$, and $N$ is a closed sub-manifold of $M$ such that $G$ action freely on $N$. 


Q: Why in a small neighborhood of $N$, $G$ also action freely?


What I do not understand is that how to let $G$ action freely on the vertical part of $ngh(N0$. 


Thanks.



Lift Lie group action on a small neighborhood
The dimension of the isotropy Lie algebra $\mathfrak g\_x$ is upper semicinuous in $x$. Thus, if it is zero in some point it is zero nearby. So, local freeness is ok as Ben has pointed out. Global freeness is not ok. One of the standard counterexamples is $G=SL(2,\mathbb C)$ acting in $3$-forms ($\cong\mathbb C^4$). Then the isotropy group of $x^2y$ is trivial but it is a group of order $3$ on a dense open subset (cyclically permuting the roots of a $3$-form). If $G$ is compact or, more generally, the action is proper then (global) freeness holds by the slice theorem.


