Let $(M,m)$ be a Riemannian manifold. In this lecture we look at the length functional and its cousin the energy functional, both defined on the path space of $M$.

We prove that the critical points of the energy functional are exactly the geodesics. We prove that the null-space of the Hessian of the energy functional can be identified with a subspace of the Jacobi fields along the geodesic.

In this lecture we used a little bit of infinite-dimensional differential geometry. Hopefully it wasn’t too scary, but if it was, don’t worry: none of it will be examinable!