= Realization of Jacobi fields by geodesic variations
Given a <Jacobi field> with initial values $J(0)=a$ and $D_tJ(0)=b$, vary the initial point along a curve with tangent $a$ and the initial velocity with covariant derivative $b$. The resulting <geodesic variation> has the same Jacobi initial values and therefore the same field by uniqueness of the linear equation. Smooth dependence on initial conditions gives a common parameter interval around any fixed compact geodesic segment, even on an incomplete manifold.
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