Jacobi field
# 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.

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

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

## Motivating example

On a unit sphere, the geodesics through the North pole are great circles. Consider two such geodesics $\gamma _{0}$ and $\gamma _{\tau }$ with natural parameter, $t\in [0,\pi ]$, separated by an angle $\tau$. The geodesic distance

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

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 $\tau$.

Instead, we can consider the derivative with respect to $\tau$ at $\tau =0$:

    ${\frac {\partial }{\partial \tau }}{\bigg |}_{\tau =0}d(\gamma _{0}(t),\gamma _{\tau }(t))=|J(t)|=\sin t.$

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

Let $e_{1}(0)={\dot {\gamma }}(0)/|{\dot {\gamma }}(0)|$ and complete this to get an orthonormal basis ${\big \e_{i}(0){\big \}}$ at $T_{\gamma (0)}M$. Parallel transport it to get a basis $\{e_{i}(t)\}$ all along $\gamma$. This gives an orthonormal basis with $e_{1}(t)={\dot {\gamma }}(t)/|{\dot {\gamma }}(t)|$. The Jacobi field can be written in co-ordinates in terms of this basis as $J(t)=y^{k}(t)e_{k}(t)$ and thus

    ${\frac {D}{dt}}J=\sum _{k}{\frac {dy^{k}}{dt}}e_{k}(t),\quad {\frac {D^{2}}{dt^{2}}}J=\sum _{k}{\frac {d^{2}y^{k}}{dt^{2}}}e_{k}(t),$

and the Jacobi equation can be rewritten as a system

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

for each $k$. This way we get a linear ordinary differential equation (ODE). Since this ODE has smooth coefficients we have that solutions exist for all $t$ and are unique, given $y^{k}(0)$ and ${y^{k}}'(0)$, for all $k$.

## Examples

Consider a geodesic $\gamma (t)$ with parallel orthonormal frame $e_{i}(t)$, $e_{1}(t)={\dot {\gamma }}(t)/|{\dot {\gamma }}|$, constructed as above.

* The vector fields along $\gamma$ given by ${\dot {\gamma }}(t)$ and $t{\dot {\gamma }}(t)$ are Jacobi fields.
* In Euclidean space (as well as for spaces of constant zero sectional curvature) Jacobi fields are simply those fields linear in $t$.
* For Riemannian manifolds of constant negative sectional curvature $-k^{2}$, any Jacobi field is a linear combination of ${\dot {\gamma }}(t)$, $t{\dot {\gamma }}(t)$ and $\exp(\pm kt)e_{i}(t)$, where $i>1$.
* For Riemannian manifolds of constant positive sectional curvature $k^{2}$, any Jacobi field is a linear combination of ${\dot {\gamma }}(t)$, $t{\dot {\gamma }}(t)$, $\sin(kt)e_{i}(t)$ and $\cos(kt)e_{i}(t)$, where $i>1$.
* The restriction of a Killing vector field to a geodesic is a Jacobi field in any Riemannian manifold.
