= Exponential and oscillatory geodesic deviation
{title2=$D^2S=\kappa S$}
In a <parallel-propagated orthonormal frame>, orthogonal <geodesic deviation> in <constant sectional curvature> reduces to three scalar equations $\ddot S^i=\kappa S^i$. For $\kappa>0$, $S=Ae^{\sqrt\kappa\tau}+Be^{-\sqrt\kappa\tau}$: generic initial data grow exponentially, but the pure decaying mode is a counterexample to universal growth. Initially comoving neighbors have $S=S(0)\cosh(\sqrt\kappa\tau)$. For $\kappa<0$, $S=A\cos(\sqrt{-\kappa}\tau)+B\sin(\sqrt{-\kappa}\tau)$, so the deviation is bounded and oscillatory. Simultaneous focusing requires compatible vector initial data and is not automatic for every oscillatory solution.
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