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Suppose $X$ is a vector field on a smooth manifold $M$. Suppose the flow $\phi : \mathcal{D} \rightarrow M $ of $X$ , where $\mathcal{D} = \{ (t,p) \in \mathbb{R} \times M : a_p < t < b_p \}$ is defined on $(-\epsilon , \epsilon)$ for $\epsilon > 0$. $(a_p, b_p)$ is the domain of the integral curve.

Then how do I show that both $\mathcal{D} = \mathbb{R} \times M$ and $X$ is a complete vector field ?

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The flow property tells you that for $t,s>0$, the curve $\phi(\phi(x,t),s)$ extends the integral curve starting at $x$ up to time $t+s$. This easily implies that if $(-\epsilon,\epsilon)\times M\subset\mathcal D$, then also $(-2\epsilon,2\epsilon)\times M\subset\mathcal D$ and inductively you get $\mathbb R\times M\subset\mathcal D$, which proves both claims.

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