Jacobi Field

A Jacobi field along a geodesic is the infinitesimal variation field of a smooth variation through geodesics. It satisfies the linear differential equation

where is the Riemann tensor with convention . A nonzero Jacobi field vanishing at two distinct times characterizes conjugate points along the geodesic. In Euclidean space, parallel coordinates give , so no nonzero Jacobi field vanishes at two times. On the unit sphere with , along a unit-speed geodesic, a normal Jacobi field is , where is a nonzero parallel tangent vector field perpendicular to the geodesic's velocity. It vanishes at and , giving conjugate points at distance .