Connections and curvature

Question

I have data defined on $\mathbb{R}^3 \times S^2$, i.e. $S(x,y,z,\theta,\phi)$. For any two points $p$ and $p'$ in $\mathbb{R}^3$ I also have a rotation $R_{pp'} \in SO(3)$ that best aligns the data on the sphere from those two points. Is there some sort of curvature (or equivalent) that can measure "the strength" of much I have rotated? Perhaps there can be a connection defined from that rotation from which curvature can be calculated, I have heard about $SU(2)$ connections from physics but not sure if it applies here.