While attempting to introduce the $\exp$ map, one is often lead through integral curves (say $\gamma:\mathbb R \times \mathfrak g \times G \rightarrow G$ ) on a Lie group $G$, with initial conditions at $t=0$ being $\dot\gamma(0)=X \in \mathfrak g$ and the initial point at the identity $e \in G$.
To obtain an integral curve, we use left invariance to send the single vector $X$ to any other required point, thus obtaining a vector field by using the group structure (see for ex: Fulton-Harris, beginning of $\S$ 8.3)
$$v_X(g)={L_g}_* X \tag{1}\label{vecf}$$
Finally, existence and uniqueness from theory of differential equations is used to declare $\gamma(t)$ as solution of
$$\dot\gamma = v_X(\gamma(t)) \tag{2}\label{intcurv}$$
My confusions are the following:
- To solve \eqref{intcurv}, it seems that i would need to know $\gamma(t)$ beforehand to be able to use $\eqref{vecf}$.
- At a formal level, is it allowed to deduce existence of $\gamma(t)$ from \eqref{intcurv} with $v_X$ being defined only implicitly?
I wanted to add that usually the presentation of $\exp$ map comes after this, defined through $\gamma $ (for ex: Fulton-Harris). Hence i would not like to use $\exp$ here.