= Sine index-form bound for positive Ricci curvature
{title2=$\sum_{i=1}^{n-1}I(V_i,V_i)\leq\frac{n-1}{2}(\pi^2/L-\kappa L)$}
Along a unit-speed <minimizing geodesic> of positive length $L$, take perpendicular parallel orthonormal fields $E_i$ and set $V_i=\sin(\pi t/L)E_i$. The <second variation of geodesic energy> makes each index form nonnegative, while $\operatorname{Ric}\geq(n-1)\kappa g$ bounds their sum as displayed. For $n\geq2$ and $\kappa>0$, lengths greater than $\pi/\sqrt\kappa$ are impossible. Completeness is what supplies a <minimizing geodesic> between arbitrary points in the <Bonnet-Myers theorem>.
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