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.

[edit] 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.

[edit] Motivating example

On a 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.

[edit] 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

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

and the Jacobi equation can be rewritten as a system

$\frac{d^2y^k}{dt^2}+|\dot\gamma|^2\sum_j y^j(t)\langle R(e_j(t),e_1(t))e_1(t),e_k(t)\rangle=0$

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 $k$ .

[edit] 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.
The Jacobi fields correspond to the geodesics on the tangent bundle (with respect to the metric on $TM$ induced by the metric on $M$ ).