# Jacobi field

In Riemannian geometry, a Jacobi field is a vector field along a geodesic $\gamma$ 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.

## Definitions and properties

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

$J(t)=\left.{\frac {\partial \gamma _{\tau }(t)}{\partial \tau }}\right|_{\tau =0}$ is a Jacobi field, and describes the behavior of the geodesics in an infinitesimal neighborhood of a given geodesic $\gamma$ .

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

${\frac {D^{2}}{dt^{2}}}J(t)+R(J(t),{\dot {\gamma }}(t)){\dot {\gamma }}(t)=0,$ where D denotes the covariant derivative with respect to the Levi-Civita connection, R the Riemann curvature tensor, ${\dot {\gamma }}(t)=d\gamma (t)/dt$ 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 $\gamma _{\tau }$ describing the field (as in the preceding paragraph).

The Jacobi equation is a linear, second order ordinary differential equation; in particular, values of $J$ and ${\frac {D}{dt}}J$ at one point of $\gamma$ 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.

## Motivating example

$d(\gamma _{0}(t),\gamma _{\tau }(t))\,$ is

$d(\gamma _{0}(t),\gamma _{\tau }(t))=\sin ^{-1}{\bigg (}\sin t\sin \tau {\sqrt {1+\cos ^{2}t\tan ^{2}(\tau /2)}}{\bigg )}.$ Computing this requires knowing the geodesics. The most interesting information is just that

$d(\gamma _{0}(\pi ),\gamma _{\tau }(\pi ))=0\,$ , for any $\tau$ .
${\frac {\partial }{\partial \tau }}{\bigg |}_{\tau =0}d(\gamma _{0}(t),\gamma _{\tau }(t))=|J(t)|=\sin t.$ Notice that we still detect the intersection of the geodesics at $t=\pi$ . Notice further that to calculate this derivative we do not actually need to know

$d(\gamma _{0}(t),\gamma _{\tau }(t))\,$ ,

rather, all we need do is solve the equation

$y''+y=0\,$ ,

for some given initial data.

Jacobi fields give a natural generalization of this phenomenon to arbitrary Riemannian manifolds.

## Solving the Jacobi equation

${\frac {D}{dt}}J=\sum _{k}{\frac {dy^{k}}{dt}}e_{k}(t),\quad {\frac {D^{2}}{dt^{2}}}J=\sum _{k}{\frac {d^{2}y^{k}}{dt^{2}}}e_{k}(t),$ and the Jacobi equation can be rewritten as a system

${\frac {d^{2}y^{k}}{dt^{2}}}+|{\dot {\gamma }}|^{2}\sum _{j}y^{j}(t)\langle R(e_{j}(t),e_{1}(t))e_{1}(t),e_{k}(t)\rangle =0$ 