I would appreciate some help showing the following statement.
Let $\omega: [0,1] \rightarrow S^2$ be a smooth curve with velocity vector $V = \omega'$, speed $v = |V|$ and Frenet frame $\{T,N\}$.
Then for all vector fields $W: S^2 \rightarrow TS^2$ along the curve $\omega$ we have:
$$R(W,V)V = v^2 \langle W , N \rangle N $$
where $R$ is the Riemannian curvature tensor on $S^2$.
If $\omega$ were arc-length parametrized / unit-length, I suppose the terms $v^2$ could join the left side because of the normalization of $N = \frac{|T'|}{T'}$, but this isn't the case here.