Editing Jacobi field (section)

==Definitions and properties==

Jacobi fields can be obtained in the following way: Take a [[smooth function|smooth]] one parameter family of geodesics <math>\gamma_\tau</math> with <math>\gamma_0=\gamma</math>, then 
:<math>J(t)=\left.\frac{\partial\gamma_\tau(t)}{\partial \tau}\right|_{\tau=0}</math>
is a Jacobi field, and describes the behavior of the geodesics in an infinitesimal neighborhood of a 
given geodesic <math>\gamma</math>.

A vector field ''J'' along a geodesic <math>\gamma</math> is said to be a '''Jacobi field''' 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 [[covariant derivative]] with respect to the [[Levi-Civita connection]], ''R'' the [[Riemann curvature tensor]], <math>\dot\gamma(t)=d\gamma(t)/dt</math> the tangent vector field, and ''t'' is the parameter of the geodesic.
On a [[Complete space|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 differential equation|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> 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 <math>\dot\gamma(t)</math> and <math>t\dot\gamma(t)</math>. These correspond respectively to the following families of reparametrizations: <math>\gamma_\tau(t)=\gamma(\tau+t)</math> and <math>\gamma_\tau(t)=\gamma((1+\tau)t)</math>.

Any Jacobi 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.