= Jacobi fields in a locally symmetric Riemannian manifold
{c}
{title2=$D_t^2J+\mathcal R_{\dot\gamma}J=0$}
Diagonalize the self-adjoint <Jacobi curvature operator> at the initial point and transport its orthonormal eigenbasis parallel along the <geodesic>. The <Jacobi field> equation reduces to $j_i''+\lambda_i j_i=0$. For zero initial position and initial derivative $b_i$, the solution is $b_i\sin(\sqrt{\lambda_i}t)/\sqrt{\lambda_i}$ when $\lambda_i>0$, $b_it$ when $\lambda_i=0$, and $b_i\sinh(\sqrt{-\lambda_i}t)/\sqrt{-\lambda_i}$ when $\lambda_i<0$. Only positive eigenvalues contribute <conjugate points>, at $t=k\pi/\sqrt{\lambda_i}$. At a common zero the multiplicity is the sum of the contributing eigenspace dimensions. The displayed convention uses the curvature operator positive on a positively curved plane; reversing the tensor convention also reverses its tensor formula.
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