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 1 If 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.