Jacobi field

In Riemannian geometry, a Jacobi field is a vector field along a geodesic ${\displaystyle \gamma }$ in a Riemannian manifold describing the difference between the geodesic and an "infinitesimally close" geodesic. In other words, the Jacobi fields along a geodesic form the tangent space to the geodesic in the space of all geodesics. They are named after Carl Jacobi.

Jacobi fields can be obtained in the following way: Take a smooth one parameter family of geodesics ${\displaystyle \gamma _{\tau }}$ with ${\displaystyle \gamma _{0}=\gamma }$, then

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