Editing Jacobi field (section)

==Examples==
Consider a geodesic <math>\gamma(t)</math> with parallel orthonormal frame <math>e_i(t)</math>, <math>e_1(t)=\dot\gamma(t)/|\dot\gamma|</math>, constructed as above.
* The vector fields along <math>\gamma</math> given by <math>\dot \gamma(t)</math> and <math>t\dot \gamma(t)</math> are Jacobi fields.
* In Euclidean space (as well as for spaces of constant zero [[sectional curvature]]) Jacobi fields are simply those fields linear in <math>t</math>.
*For Riemannian manifolds of constant negative sectional 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 sectional 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>.
*The restriction of a [[Killing vector field]] to a geodesic is a Jacobi field in any Riemannian manifold.