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Let a (connected) Lie group $G$ act on $M$. Assume that the action is locally free. (In other words, if the fundamental vector field of $X \in \mathrm{Lie(G)}$ $$ \underline{X}(p) := \left.\dfrac{d}{dt} \exp(tX) \cdot p\right|_{t=0} $$ vanishes at some point $p \in M$, then $X=0$.)

Let $K$ be a Lie subgroup of $G$ and $N$ a $K$-invariant submanifold of $M$. We suppose the following condition.

$$ q \in N,\ g \in G,\ gq \in N \implies g \in K. $$

Question: If $\underline{X}(q) \in T_qN$, then does $X$ belong to $\mathrm{Lie}(K)$?

(For example, we may take $G:=\mathbb{C}^*=:M$ and $K:=S^1=:N$. Here the action of $G$ on $M$ is given by complex multiplications. But if we take $M=\mathbb{C}$ and $N=\{0\}$, then the statement does not hold.)

We may assume $G$ and $K$ are compact, if we need.

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