It is of interest to consider functions on the space of curves , where is an interval and is a smooth manifold. To study maxima and minima, it is of interest to consider variations of curves, holding the endpoints fixed. Let be smooth. A variation of is a smooth map
with , and for all . For a variation of , define the variation vector field (which is an analog of a “tangent vector”)
this is a vector field along . Similarly we can define the “velocity vector field” along . If is provided with a connection, we can define the “acceleration vector field” , where denotes covariant differentiation.
Given a vector field along , we can construct a variation of with as the variation vector field: take .
Variations of geodesics and Jacobi fields Let now be a manifold with a symmetric connection .Let be a variation of a geodesic such that for any , is a geodesic as well. Then the variation vector field satisfies a certain differential equation. Now
We have used the symmetry of . Now we can write this as
By geodesy, the first part vanishes, and the second is
We have shown that satisfies the Jacobi equation
Any vector field along satisfying this is called a Jacobi field.
The differential of the exponential map Let , and consider the exponential map where is a neighborhood of the origin in . Let . Now the map takes horizontal lines to geodesics in when are small enough. This can be viewed as a vector field along the geodesic . The variation vector field is thus a Jacobi field, and also at is Note that satisfies and
Proposition 1 Suppose are sufficiently small. Let be the Jacobi field along the geodesic with (i.e. using the ODE theorems). Then
I will next explain how to use this fact to prove the Cartan-Hadamard theorem on manifolds of negative curvature.
Source: These notes, from an introductory course on differential geometry.