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