Locally symmetric Riemannian manifold ($\nabla R=0$)

A Riemannian manifold is locally symmetric when its Riemann curvature tensor is parallel for the Levi-Civita connection. Along a geodesic, parallel transport then conjugates the Jacobi curvature operator at the initial point to the operator at every later point. Thus its eigenvalues stay constant. The condition is local and does not assert that global point reflections extend to isometries of the whole manifold.

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 $j_i^{\prime \prime }+{\lambda }_i{j}_i=0$. For zero initial position and initial derivative $b_i$, the solution is $b_i\sin (\sqrt{{\lambda }_i}t)/\sqrt{{\lambda }_i}$ when ${\lambda }_i>0$, $b_{i}t$ 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.