I’m going to keep the same notation as before. In particular, we’re studying how the energy integral behaves with respect to variations of curves. Now I want to prove the second variation formula when **is a geodesic**.Now to compute , for further usage. We already showed Differentiating again yields the messy formula for :

Call these .

** **

Now is the easiest, since by symmetry of the Levi-Civita connection we get For vector fields along with , we have This is essentially a forum of integration by parts. Indeed, the difference between the two terms is

So if we plug this in we get

** **

Next, we can write

Now measures the difference from commutation of . In particular this equals

By antisymmetry of the curvature tensor (twice!) the second term becomes

Now we look at the first term, which we can write as

since . But this is clearly zero because is constant on the vertical lines . If we put everything together we obtain the following “second variation formula:”

Theorem 1If is a geodesic, then

Evidently that was some tedious work, and the question arises: Why does all this matter? The next goal is to use this to show when a geodesic **cannot** minimize the energy integral—which means, in particular, that it doesn’t minimize length. Then we will obtain global comparison-theoretic results.