Jacobi fields and geodesic variation

Let F(s,t) be a smooth variation whose t-curves are affinely parameterized geodesics. Along the central curve write T for its velocity and J for the s-variation field. The coordinate fields in the parameter rectangle commute, and torsion freedom gives $\nabla_sT=\nabla_tJ$. Differentiate the geodesic identity $\nabla_tT=0$ in the s-direction and commute the covariant derivatives using the curvature convention from the preceding lesson. This produces the Jacobi equation displayed below.

The derivative D/dt acts along the moving tangent spaces; replacing it by ordinary coordinate differentiation omits connection terms. Initial values J(0) and DJ/dt(0) determine a unique solution of this linear second-order equation along a given geodesic. Conversely, locally every Jacobi field can be realized by varying initial geodesic data. Tangential fields of the form (a+bt)T arise from changing the starting parameter or speed, so not every Jacobi field represents normal spatial focusing.

On a unit-speed geodesic in a constant-sectional-curvature space of dimension at least two, choose a unit parallel normal field E. A field J=jE then satisfies the scalar equation $j''+Kj=0$. With initial conditions j(0)=0 and j'(0)=1, positive curvature gives a sine, zero curvature gives linear separation, and negative curvature gives hyperbolic sine. The normalization of initial derivative is important: changing it scales the entire field and changes its interpretation as a variation of initial direction.

These formulas are exact model-space calculations, not a replacement for a general curvature-varying matrix equation. The parallel normal frame also matters; a rotating frame introduces extra derivative terms. In positive curvature the sine first returns to zero at π divided by the square root of K. This is infinitesimal refocusing of distinct initial directions. In negative curvature, the chosen normal solution grows faster than linear and has no second zero after the initial point.

A point at positive parameter time along a geodesic is conjugate to its starting point if a nonzero Jacobi field vanishes at both times. For variations fixing the starting point, this corresponds to a nontrivial kernel in the differential of the exponential map at the appropriate initial velocity. Conjugacy concerns infinitesimal initial-data degeneracy. Distinct finite geodesic routes can also meet without such a differential singularity, so endpoint nonuniqueness is a broader phenomenon.

The second variation relates Jacobi fields to minimizing behavior: a geodesic segment cannot remain minimizing after an interior conjugate point. At an endpoint conjugate point, it may still minimize, as an antipodal great-circle arc on a sphere does. A cut point can occur earlier because a competing minimizing route appears. The flat cylinder later supplies exactly that distinction: its topology can create a cut point even though its zero-curvature normal Jacobi fields do not refocus.