Jacobi fields form the radical of the fixed-endpoint index form

On continuous piecewise smooth vector field along a curve with zero endpoints, the radical of a bilinear form of the Riemannian index form is exactly the space of endpoint-vanishing Jacobi fields. Integration by parts against tests supported inside each smooth piece gives the Jacobi equation; tests with arbitrary values at breakpoints force continuity of the covariant derivative, hence a global smooth Jacobi field. This characterization uses $I(V,W)=0$ for every test field $W$, not merely $I(V,V)=0$, which is insufficient for an indefinite form.

// Target: riemannian-geometry.bigb