Jacobi field

Let $(M,g)$ be a Riemannian manifold, let $I$ be a nondegenerate interval, and let $\gamma :I\to M$ be an affinely parametrized geodesic. For a smooth vector field $J$ along $\gamma$, write $D_{t}J$ for its covariant derivative along $\gamma$ and $D_t^{2}J=D_t(D_{t}J)$, using Covariant derivative along a curve. The curvature operator $R$ has the convention $R(X,Y)Z={\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}Z-{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_{X}Z-{\mathrm{\nabla }}_{[X,Y]}Z,$ as fixed by Covariant derivatives commute up to curvature in a two parameter variation and used in Riemann curvature four-tensor. A smooth field $J$ is a Jacobi field along $\gamma$ when it satisfies the Jacobi equation $D_t^{2}J+R(J,\dot{\gamma })\dot{\gamma }=0$ throughout $I$. We write $\mathcal{J}(\gamma )$ for the set of these fields.

The interval is required to have nonempty interior so that the covariant derivatives are defined; if an endpoint is included, the equation there uses the one-sided derivatives of the smooth field. Constant geodesics are included: when $\dot{\gamma }=0$, the curvature term vanishes and the equation is $D_t^{2}J=0$. On a zero-dimensional manifold the only field is zero, so the equation holds. In dimension one the same equation and curvature convention apply without a separate restriction. The definition classifies a specified field and geodesic and makes no existence choice.