My question specifies on that the left translation of a left invariant vector field on a Lie group $G$ can be seen as a parallel transport along all possible curves. How can I see that this is true?
I found a result saying that the vector field $V=V^\alpha e_{\alpha}$ is left invariant if and only if $V^\alpha$ is constant with respect to any left invariant frames $e_\alpha$. I cant see how this follows from the definition
The definition of Left-invariant vector field is given as follows:
Definition: A vector field $V$ on a Lie group G is left invariant if $$\textrm dL_gV(x)=X(L_g(x))=X(gx)$$ with $x,g\in G$.
I hoped this result and the fact that $G$ is parallelizable (this will give me the trivial connection if we use the left invariant frame $e_\alpha$ and the covariant derivative along a curve $\gamma(t)$ is then $V'(\gamma(t))$ ) will give that the covariant derivative vainishes. Is this reasoning correct?
Thank you for any help