51. The length and energy functionals of a Riemannian manifold
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!
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