If $\exp: T_p(G) \rightarrow G$ is the expoenential map of a lie group, then what does the adjoint operator (as in $\langle Ax,y\rangle=\langle x,A^*,y\rangle$) of the derivative of exp look like?
i.e: $d_v\exp_p(v)$ and $d_p\exp_p(v)$?
Is is Related to ad?
(Sorry if this is an obvious question but I've always wondered).
For example: on the before last page in this paper the authors derive an expression for these, do these arise from $S^n$'s Lie structure?