Jacobi field

In Riemannian geometry, a Jacobi field is a vector field along a geodesic formula_1 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.
Definitions and properties.
Jacobi fields can be obtained in the following way: Take a smooth one parameter family of geodesics formula_2 with formula_3, then 
is a Jacobi field, and describes the behavior of the geodesics in an infinitesimal neighborhood of a 
given geodesic formula_1.
A vector field "J" along a geodesic formula_1 is said to be a Jacobi field if it satisfies the Jacobi equation:
where "D" denotes the covariant derivative with respect to the Levi-Civita connection, "R" the Riemann curvature tensor, formula_8 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 formula_2 describing the field (as in the preceding paragraph).
The Jacobi equation is a linear, second order ordinary differential equation;
in particular, values of formula_10 and formula_11 at one point of formula_1 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 formula_13 and formula_14. These correspond respectively to the following families of reparametrisations: formula_15 and formula_16.
Any Jacobi field formula_10 can be represented in a unique way as a sum formula_18, where formula_19 is a linear combination of trivial Jacobi fields and formula_20 is orthogonal to formula_13, for all formula_22. 
The field formula_23 then corresponds to the same variation of geodesics as formula_10, only with changed parameterizations.
Motivating example.
On a sphere, the geodesics through the North pole are great circles. Consider two such geodesics formula_25 and formula_2 with natural parameter, formula_27, separated by an angle formula_28. The geodesic distance 
is 
Computing this requires knowing the geodesics. The most interesting information is just that
Instead, we can consider the derivative with respect to formula_28 at formula_34:
Notice that we still detect the intersection of the geodesics at formula_36. Notice further that to calculate this derivative we do not actually need to know 
rather, all we need do is solve the equation 
for some given initial data. 
Jacobi fields give a natural generalization of this phenomenon to arbitrary Riemannian manifolds.
Solving the Jacobi equation.
Let formula_39 and complete this to get an orthonormal basis formula_40 at formula_41. Parallel transport it to get a basis formula_42 all along formula_1. 
This gives an orthonormal basis with formula_44. The Jacobi field can be written in co-ordinates in terms of this basis as formula_45 and thus
and the Jacobi equation can be rewritten as a system
for each formula_48. This way we get a linear ordinary differential equation (ODE). 
Since this ODE has smooth coefficients we have that solutions exist for all formula_22 and are unique, given formula_50 and formula_51, for all formula_48.
Examples.
Consider a geodesic formula_53 with parallel orthonormal frame formula_54, formula_55, constructed as above.