How can one define parallel transport on lie groups?

Question

I want to transport a tangent vector $T_p$ at a point $p$ to some other point $q$ on a lie group. Do I have to use $log(p^{-1}q)$, which is also a tangent vector, to achieve the same?

Answer 1

Let $G$ be a Lie group. For all $g,h\in G$ there is a canonical linear isomorphism $T_gG\to T_hG$ defined as follows. For $a\in G$ let $L_a:G\to G$ be left translation by $a$, i.e. $L_a(b)=ab$. Specializing with $a=hg^{-1}$ and taking the derivative of $L_a$ we get a linear map $$dL_{hg^{-1}}:T_gG\to T_hG.$$ This is an isomorphism since $dL_{g^{-1}h}$ is an inverse.

Note: Using these facts one can show that Lie groups are parallelizable meaning that $TG$ is isomorphic to the trivial bundle $G\times\mathfrak{g}$, where $\mathfrak{g}=T_eG$.