Conjugate-point criterion for the Riemannian index form

On continuous piecewise smooth perpendicular fields with zero endpoints, the Riemannian index form is nonnegative exactly when no point strictly inside the geodesic segment is conjugate to its initial point. It is positive on every nonzero field exactly when no conjugate point occurs up to and including the final endpoint. At a first conjugate endpoint its nullspace consists of the endpoint-vanishing Jacobi fields. These statements distinguish nonnegative quadratic forms from strictly positive ones and do not assume global minimization.

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 $P(a)=-D_{t}J(a)$; integration by parts gives $I(Z,P)=-|D_{t}J(a){|}^2<0$. Thus $I(Z+\epsilon P,Z+\epsilon P)<0$ for small positive $\epsilon$. 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