Suppose I have a manifold $M$ , a compact submanifold $Q$ and a time-dependent vector field $X_t$ such that $X_t(x)=0$ for every $x\in Q$. Now I would like to understand why there is a small enough neighborhood $U$ of $Q$ where we can integrate the vector field to $\phi_t:U\times [0,1]\rightarrow U$ is well-defined.
I get that we will that $\phi_t$ is defined for all time in $Q$ since it's just constant do to the fact that $X_t(x)=0$ for $x\in Q$. Locally around each point I can also find an $\epsilon>0$ such that the flow is defined but this may fail to be $[0,1]$ so I don't think this helps. And so I am out of ideas of what one can use. I mean I can cover $Q$ with neighborhoods $V_1,...,V_n$ where I have the flow defined for time $\epsilon_1,...,\epsilon_n$, since $Q$ is compact, and so I can go around this times until I equal $1$?
Do you have any suggestions ? Thanks in advance.