Lemma 1 If is a complete Riemannian manifold of dimension and Ricci curvature bounded below by , then a geodesic of length does not minimize energy if .
If we choose a geodesic parametrized by unit length, we then will be done if we find an endpoint-preserving variation of with . In other words if we can find a vector field along with and
we will obtain a contradiction; cf. the initial discussion in yesterday’s post and the second variation formula.
Choose parallel vector fields along which, together with , form an orthonormal basis for at each . To do this choose for , and then parallel translate. Then since is skew-symmetric in the last two variables and
In particular, we could use the sum of the to get the vector field satisfying the boxed inequality—except . So we define
By the product rule for covariant differentiation and the paralellism of ,
So using orthonormality and , we can get that
if minimizes energy, so .
This ends my MaBloWriMo series on differential geometry; there’s one more day in November, of course, but I need to first learn more new material before I can go further. Besides, it is probably healthy both for this blog and for myself to cover some other topics.
The posting over the next few weeks will probably be less structured than these entries. I’m not yet quite sure what I want to discuss, but likely the topic will be one or two out of Riemann surfaces, Koszul complexes and depth, spectral sequences, and singular integrals. Feel free to suggest something.