Vector Fields Question 4

Question

I am struggling with the following question:

Prove that any left invariant vector field on a Lie group is complete.

Any help would be great!

Answer 1

Call your lie group $G$ and your vector field $V$.

It suffices to show that there exists $\epsilon > 0$ such that given $g \in G$ (notice the order of the quanitfiers!) there exists an integral curve $\gamma_g : (-\epsilon, \epsilon) \rightarrow G$ with $\gamma_g (0) = g$, I.e. a curve $\gamma$ starting at $p$ with $\gamma_\ast (\frac{d}{dt}|_s) = V_{\gamma(s)}$.

Now there exists an integral curve $\gamma _e$through the identity, $e$, defined on some open neighborhood $(-\epsilon, \epsilon)$ of $0$. This is the $\epsilon$ we choose. For any $g \in G$ we use the left invariance of $V$ to check that $L_g \circ \gamma_e$ is an integral curve: $L_{g_\ast} (\gamma _{e_\ast} (\frac{d}{dt}|_s) = L_{g_\ast} ( V_{\gamma_e (s)}) = V_{L_g \circ \gamma_e (s)}$. This completes the proof.