Jacobi field

In Riemannian geometry, a Jacobi field is a certain type of vector field along a geodesic $γ$ in a Riemannian manifold. Jacobi fields are one of the basic objects of study in Riemannian geometry; for the origin of the name, see Carl Jacobi.

Contents

1 Definitions and properties

2 Motivating example

3 Solving the Jacobi Equation

4 Examples

5 References

Definitions and properties

Jacobi fields can be obtained in the following way: Take a smooth one parameter family of geodesics $γ τ$ with $γ 0 = γ$ , then

$J(t)=\partial\gamma_\tau(t)/\partial \tau|_{\tau=0}$

is a Jacobi field.

A field J is a Jacobi field if and only 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 Levi-Civita connection, R the curvature tensor and $\dot\gamma(t)=d\gamma(t)/dt$ . On a complete Riemannian manifold, for any Jacobi field there is a family of geodesics $γ τ$ 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 $γ$ define uniquely the Jacobi field. Further, the sum of Jacobi fields on a given geodesic is again a Jacobi field.

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: $γ τ (t) = γ(τ + t)$ and $γ τ (t) = γ((1 + τ) t)$ .

Any Jacobi field 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 parametrizations.

Motivating example

On a sphere, the geodesics through the North pole are great circles. Consider two such geodesics $γ 0$ and $γ τ$ with natural parameter, $t\in [0,\pi]$ , separated by an angle $τ$ . The geodesic distance $d (γ 0 (t),γ τ (t))$ 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 (γ 0 (π),γ τ (π)) = 0

, for any

τ

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

$\frac{\partial}{\partial\tau}\bigg|_{\tau=0}d(\gamma_0(t),\gamma_\tau(t))=|J(t)|=\sin t.$

Notice that we still detect the intersection of the geodesics at $t = π$ . Notice further that to calculate this derivative we do not actually need to know $d (γ 0 (t),γ τ (t))$ , 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 $T γ(0) M$ . Parallel transport it to get a basis ${e i (t)}$ all along $γ$ . This gives an orthonormal basis with $e_1(t)=\dot\gamma(t)/|\dot\gamma(t)|$ . The Jacobi field is $J (t) = y k (t) e k (t)$ 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 $k$ .

Examples

Consider a geodesic $γ(t)$ with parallel basis frame $e i (t)$ , $e_1(t)=\dot\gamma(t)/|\dot\gamma|$ , constructed as above.

In Euclidean space (as well as for spaces of constant zero curvature) Jacobi fields are simply those fields linear in $t$ .

For Riemannian manifolds of constant negative 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 curvature $k 2$ , any Jacobi field is a linear combination of $\dot\gamma(t)$ , $t\dot\gamma(t)$ , $sin(k t) e i (t)$ and $cos(k t) e i (t)$ , where $i > 1$ .

References

[do Carmo] M. P. do Carmo, Riemannian Geometry, Universitext, 1992.