Let $f:SO(3) \rightarrow \mathbb{R}$, $\tilde{R}:\mathbb{R}^2\rightarrow SO(3)$, and $\bar{f}:\mathbb{R}^2\rightarrow\mathbb{R}:q\mapsto f(\tilde{R}(q))$.
I am interested in computing the second derivative of $\bar{f}$.
The first derivative of $\bar{f}$ has a nice structure owing to how derivatives work on $SO(3)$, since tangent spaces of $SO(3)$ are isomorphic to $so(3)$.
$\text{D}\bar{f}(q)\cdot v = \text{D}f(\tilde{R}(q))\cdot(\tilde{R}(q)\tilde{R}(q)^\top\text{D}\tilde{R}(q)\cdot v)=\text{D}f(\tilde{R}(q))\cdot(\tilde{R}(q)(J(q)v)^\hat{})$
where $a\hat{}$ is the antisymmetric matrix corresponding to the vector $a$. The above formula will give a nice formula for $\text{D}\bar{f}(q):\mathbb{R}^2\rightarrow\mathbb{R}^2$.
I am wondering if it is possible to take advantage of the special structure of $SO(3)$ in computing the second derivative.
I am not familiar with connections on Lie groups such as $SO(3)$, but I am guessing there is some relevance, and would appreciate any pointers.
Thank you.