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

is a Jacobi field, and describes the behavior of the geodesics in an infinitesimal neighborhood of a given geodesic ${\displaystyle \gamma }$.

A vector field J along a geodesic ${\displaystyle \gamma }$ is said to be a Jacobi field if it satisfies the Jacobi equation:

where D denotes the covariant derivative with respect to the Levi-Civita connection, R the Riemann curvature tensor, ${\displaystyle {\dot {\gamma }}(t)=d\gamma (t)/dt}$ the tangent vector field, and t is the parameter of the geodesic. On a complete Riemannian manifold, for any Jacobi field there is a family of geodesics ${\displaystyle \gamma _{\tau }}$ describing the field (as in the preceding paragraph).

The Jacobi equation is a linear, second order ordinary differential equation; in particular, values of ${\displaystyle J}$ and ${\displaystyle {\frac {D}{dt}}J}$ at one point of ${\displaystyle \gamma }$ uniquely determine the Jacobi field. Furthermore, the set of Jacobi fields along a given geodesic forms a real vector space of dimension twice the dimension of the manifold.

As trivial examples of Jacobi fields one can consider ${\displaystyle {\dot {\gamma }}(t)}$ and ${\displaystyle t{\dot {\gamma }}(t)}$. These correspond respectively to the following families of reparametrizations: ${\displaystyle \gamma _{\tau }(t)=\gamma (\tau +t)}$ and ${\displaystyle \gamma _{\tau }(t)=\gamma ((1+\tau )t)}$.

Any Jacobi field ${\displaystyle J}$ can be represented in a unique way as a sum ${\displaystyle T+I}$, where ${\displaystyle T=a{\dot {\gamma }}(t)+bt{\dot {\gamma }}(t)}$ is a linear combination of trivial Jacobi fields and ${\displaystyle I(t)}$ is orthogonal to ${\displaystyle {\dot {\gamma }}(t)}$, for all ${\displaystyle t}$. The field ${\displaystyle I}$ then corresponds to the same variation of geodesics as ${\displaystyle J}$, only with changed parametrizations.