= Nonpositive sectional curvature excludes conjugate points
{title2=$(|J|^2)^{\prime\prime}\ge0$}
For a <Jacobi field> along a <geodesic> with nonpositive <sectional curvature>, the curvature convention with positive round-sphere curvature gives
$$
(|J|^2)''=2|D_tJ|^2-2\langle R(J,\dot\gamma)\dot\gamma,J\rangle\ge0.
$$
Thus $|J|^2$ is nonnegative and convex. If it vanishes at both ends of a segment, convexity forces it to vanish everywhere. No nonzero <Jacobi field> can vanish at both ends, so the segment has no <conjugate points>. Only curvature of planes containing the <geodesic> tangent is needed, and completeness is unnecessary.
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