= Jacobi fields vanishing at their initial point
{c}
{title2=$J(t)=(d\exp_p)_{tV}(tD_tJ(0))$}
For $\gamma(t)=\exp_p(tV)$, every <Jacobi field> with $J(0)=0$ is uniquely of the form
$$
J(t)=(d\exp_p)_{tV}(tB),\qquad B=D_tJ(0)\in T_pM.
$$
Differentiate the <geodesic variation> $\exp_p(t(V+sB))$ in $s$ to obtain the formula. Uniqueness of the Jacobi equation with prescribed initial value and derivative shows that all such fields are obtained. A nontrivial field vanishing at $t=L>0$ therefore corresponds to a nonzero vector in the kernel of $(d\exp_p)_{LV}$, giving the <conjugate point> criterion.
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