Motivating Example
On a sphere, the geodesics through the North pole are great circles. Consider two such geodesics and with natural parameter, separated by an angle . The geodesic distance
is
Computing this requires knowing the geodesics. The most interesting information is just that
- , for any .
Instead, we can consider the derivative with respect to at :
Notice that we still detect the intersection of the geodesics at . 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.
Read more about this topic: Jacobi Field
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