Sine index-form bound for positive Ricci curvature (${\sum }_{i=1}^{n-1}I(V_i,V_i)\le \frac{n-1}{2}({\pi }^2/L-\kappa L)$)

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 $\mathrm{Ric}\ge (n-1)\kappa g$ bounds their sum as displayed. For $n\ge 2$ 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.