= Loss of geodesic minimality beyond a conjugate point
Let a nonzero <Jacobi field> $J$ vanish at times zero and $a$. On a longer <geodesic> segment ending at $b>a$, extend it by zero to a continuous piecewise smooth field $Z$. Then $I(Z,Z)=0$. Choose an endpoint-vanishing field $P$ with $P(a)=-D_tJ(a)$; <integration by parts> gives $I(Z,P)=-|D_tJ(a)|^2<0$. Thus $I(Z+\varepsilon P,Z+\varepsilon P)<0$ for small positive $\varepsilon$. Smooth approximation preserves negativity. The <second variation of geodesic energy> supplies a shorter path, so a <minimizing geodesic> cannot extend beyond its first <conjugate point>.
// Target: differential-geometry.bigb
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