Jacobi field
From Academic Kids
In Riemannian geometry, a Jacobi field is a certain type of vector field along a geodesic <math>\gamma<math> 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.
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Definitions and properties
Jacobi fields can be obtained in the following way: Take a smooth one parameter family of geodesics <math>\gamma_\tau<math> with <math>\gamma_0=\gamma<math>, then
- <math>J(t)=\partial\gamma_\tau(t)/\partial \tau|_{\tau=0}<math>
is a Jacobi field.
A field J is a Jacobi field if and only if it satisfies the Jacobi equation:
- <math>\frac{D^2}{dt^2}J(t)+R(J(t),\dot\gamma(t))\dot\gamma(t)=0,<math>
where D denotes the Levi-Civita connection, R the curvature tensor and <math>\dot\gamma(t)=d\gamma(t)/dt<math>. On a complete Riemannian manifold, for any Jacobi field there is a family of geodesics <math>\gamma_\tau<math> describing the field (as in the preceding paragraph).
The Jacobi equation is a linear second order ordinary differential equation; in particular, values of <math>J<math> and <math>\frac{D}{dt}J<math> at one point of <math>\gamma<math> 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 <math>\dot\gamma(t)<math> and <math>t\dot\gamma(t)<math>. These correspond respectively to the following families of reparametrisations: <math>\gamma_\tau(t)=\gamma(\tau+t)<math> and <math>\gamma_\tau(t)=\gamma((1+\tau)t)<math>.
Any Jacobi field field <math>J<math> can be represented in a unique way as a sum <math>T+I<math>, where <math>T=a\dot\gamma(t)+bt\dot\gamma(t)<math> is a linear combination of trivial Jacobi fields and <math>I(t)<math> is orthogonal to <math>\dot\gamma(t)<math>, for all <math>t<math>. The field <math>I<math> then corresponds to the same variation of geodesics as <math>J<math>, only with changed parametrizations.
Motivating example
On a sphere, the geodesics through the North pole are great circles. Consider two such geodesics <math>\gamma_0<math> and <math>\gamma_\tau<math> with natural parameter, <math>t\in [0,\pi]<math>, separated by an angle <math>\tau<math>. The geodesic distance <math>d(\gamma_0(t),\gamma_\tau(t))<math> is
- <math>d(\gamma_0(t),\gamma_\tau(t))=\sin^{-1}\bigg(\sin t\sin\tau\sqrt{1+\cos^2 t\tan^2(\tau/2)}\bigg).<math>
Computing this requires knowing the geodesics. The most interesting information is just that
- <math>d(\gamma_0(\pi),\gamma_\tau(\pi))=0<math>, for any <math>\tau<math>.
Instead, we can consider the derivative with respect to <math>\tau<math> at <math>\tau=0<math>:
- <math>\frac{\partial}{\partial\tau}\bigg|_{\tau=0}d(\gamma_0(t),\gamma_\tau(t))=|J(t)|=\sin t.<math>
Notice that we still detect the intersection of the geodesics at <math>t=\pi<math>. Notice further that to calculate this derivative we do not actually need to know <math>d(\gamma_0(t),\gamma_\tau(t))<math>, rather, all we need do is solve the equation <math>y''+y=0<math>, for some given initial data.
Jacobi fields give a natural generalization of this phenomenon to arbitrary Riemannian manifolds.
Solving the Jacobi equation
Let <math>e_1(0)=\dot\gamma(0)/|\dot\gamma(0)|<math> and complete this to get an orthonormal basis <math>\big\{e_i(0)\big\}<math> at <math>T_{\gamma(0)}M<math>. Parallel transport it to get a basis <math>\{e_i(t)\}<math> all along <math>\gamma<math>. This gives an orthonormal basis with <math>e_1(t)=\dot\gamma(t)/|\dot\gamma(t)|<math>. The Jacobi field is <math>J(t)=y^k(t)e_k(t)<math> and thus
- <math>\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),<math>
and the Jacobi equation can be rewritten as a system
- <math>\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<math>
for each <math>k<math>. This way we get a linear ordinary differential equation (ODE). Since this ODE has smooth coefficients we have that solutions exist for all <math>t<math> and are unique, given <math>y^k(0)<math> and <math>{y^k}'(0)<math>, for all <math>k<math>.
Examples
Consider a geodesic <math>\gamma(t)<math> with parallel basis frame <math>e_i(t)<math>, <math>e_1(t)=\dot\gamma(t)/|\dot\gamma|<math>, constructed as above.
In Euclidean space (as well as for spaces of constant zero curvature) Jacobi fields are simply those fields linear in <math>t<math>.
For Riemannian manifolds of constant negative curvature <math>-k^2<math>, any Jacobi field is a linear combination of <math>\dot\gamma(t)<math>, <math>t\dot\gamma(t)<math> and <math>\exp(\pm kt)e_i(t)<math>, where <math>i>1<math>.
For Riemannian manifolds of constant positive curvature <math>k^2<math>, any Jacobi field is a linear combination of <math>\dot\gamma(t)<math>, <math>t\dot\gamma(t)<math>, <math>\sin(kt)e_i(t)<math> and <math>\cos(kt)e_i(t)<math>, where <math>i>1<math>.
References
[do Carmo] M. P. do Carmo, Riemannian Geometry, Universitext, 1992.
