Geodesic variation
= Geodesic variation
{title2=$F(s,t),\quad\nabla_t\partial_tF=0$}
A geodesic variation of a given <geodesic> is a smooth family $F(s,t)$ with $F(0,t)=\gamma(t)$, whose curves at fixed $s$ are affinely parametrized geodesics. Differentiating its geodesic equation and using the torsion-free <Levi-Civita connection> gives the <Jacobi field> equation $D_t^2J+R(J,\dot\gamma)\dot\gamma=0$ for $J=\partial_sF|_{s=0}$.