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Let $M$ be a compact oriented manifold without boundary, $G$ be a compact Lie group acting smoothly on $M$, and $\mathfrak{g}$ be its Lie algebra. $X_{M},Y_{M}$ be the vector field on $M$ corresponding with $X,Y\in\mathfrak{g}$. $\langle,\rangle$ is the Riemannian metric and we extend it to complex vector field by linear.

Let $$M_{0}=(x\in M \mid \langle X_{M}(x)+\sqrt{-1}Y_{M}(x), X_{M}(x)+\sqrt{-1}Y_{M}(x)\rangle=0)$$ Assume that $M_{0}$ be the submanifold of $M$.

I have some questions:

1.How to find the non-trivial subgroup $G_{0}$ of $G$ such that $G_{0}$ is the group action on $M_{0}$? I mean that if $\forall g\in G_{0}, \forall m\in M_{0}$ then $g(m)\in M_{0}$.

2.Let $m\in M_{0}$, $X\in\mathfrak{g}$. If we want $exp(-tX)m\in M_{0}$, so what condition we need? This $X\in\mathfrak{g}$ is the same as the $X$ to construct $M_{0}$ corresponding with $X_{M}$.

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You haven't told us what $Y_M$ is. In fact, your question seems to leave out a lot, so we probably can't help you unless you tell us more about what you are doing. –  Robert Bryant Feb 21 '13 at 15:03
    
What is the inner product? Is $M$ Riemannian? When you say Set $M_0$ be the submanifold'' do you mean Assume that $M_0$ is a submanifold''? In question 2, do you mean for all $t$? Probably you want $G_0$ to be the stabilizer of the zeroes of $X_M$, and then in question 2 clearly $X_M$ fixes every point of $M_0$, and so $exp(-tX)m=m$ for all $m \in M_0$. But some clarification of the question would be appreciated. –  Ben McKay Feb 21 '13 at 21:42
    
@ Robert Bryant: Thank you. I have edit the question. @ Ben McKay: Thank you. Yes, I assume that $M$ is Riemannian and $M_{0}$ is a submanifold in $M$. In question 2 $G_{0}$ to is not the stabilizer of the zeroes of $X_{M}$, because $M_{0}$ is not the zeroes of $X_{M}$, and $exp(-tX)m=m$ is only a special case. Here, t is small not for all. –  Chen Feb 22 '13 at 1:36

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