= Solution
Use $t$ as an <affine parameter> on each member of the <geodesic variation>, so $\nabla_TT=0$. The parameter directions commute, giving $[T,S]=0$ along the variation. The <torsion-free connection> then gives
$$
\nabla_TS-\nabla_ST=[T,S]=0.
$$
Consequently the <covariant derivative> and curvature definition give
$$
\begin{aligned}
\nabla_T\nabla_TS
&=\nabla_T\nabla_ST\\
&=R(T,S)T+\nabla_S\nabla_TT+\nabla_{[T,S]}T\\
&=R(T,S)T.
\end{aligned}
$$
This is \b[the geodesic-deviation equation with the stated curvature sign]:
$$
\boxed{\nabla_T\nabla_TS=R(T,S)T=R^\alpha{}_{\beta\gamma\delta}T^\beta T^\gamma S^\delta e_\alpha.}
$$
The component expression defines the slot order: $\beta$ is the vector acted on, and $\gamma,\delta$ are the two derivative directions of the <Riemann curvature tensor>. Both the commuting parameter directions and affine parametrization are essential; a general connecting <vector field> or a nonaffine parameter would add terms.
Back to article page
