I read somewhere that minimum energy paths are geodesics under the Levi-Civita connection, on a Riemannian energy landscape.
The Levi-Civita connection is the "unique connection on the tangent bundle of a manifold that preserves the Riemannian metric and is torsion-free".
What exactly is the intuition of geodesics under the Levi-Civita connection? Why would torsion-free connections give geodesics with least energy?
Best Answer
If $\nabla$ is any connection and $f$ a function, its Hessian with respect to $\nabla$ is $\mathrm{Hess}^{\nabla}f = \nabla \mathrm{d}f$, and one can see, after a messy calculation, that: $$ \mathrm{Hess}^{\nabla}f(X,Y) - \mathrm{Hess}^{\nabla}f(Y,X) = \pm\mathrm{d}f\left([X,Y] - (\nabla_XY - \nabla_YX) \right) $$ (where the $\pm$ sign is here because I don't remember the exact sign, but the computations are not that hard, just messy.) Hence, Hessians are symmetric if and only if the connection is torsion-free. This is the main motivation to consider torsion-free connections: in the euclidean space, Hessians are symmetric!
Moreover, the fundamental theorem of Riemannian geometry tells us that on a Riemannian manifold, there is a unique connexion that is torsion-free and lets the metric invariant, that is: $$ \forall X,Y,Z, \left(\nabla_Zg\right)(X,Y) = Z\cdot g\left(X,Y \right) - g\left(\nabla_ZX,Y\right) - g\left(X,\nabla_ZY\right) = 0. $$ (compare with the euclidean case, where $\langle X,Y\rangle ' = \langle X',Y\rangle + \langle X, Y' \rangle$.) This theorem thus says that given any Riemannian metric $g$, there is a connection that is better than others: Hessians are symmetric and the metric is invariant under the action. We call it the Levi-Civita connexion.
If a connection is chosen, a geodesic is a parametrized curve satisfying the equation of geodesics : $\nabla_{\gamma'}\gamma' = 0$. Thus a curve $\gamma$ is a geodesic with respect to the connection, and can be a geodesic for some connection $\nabla^1$ but not for another connecion $\nabla^2$. Therefore, your question does not really have sense: we do not say that a connexion gives the least energy of a geodesic. I think you got confused, believing that being a geodesic is an intrinsic notion, but it really depends on the connection you consider.
Now, suppose $(M,g)$ is a Riemannian manifold endowed with its Levi-Civita connexion. Then if $\gamma : [a,b] \to M$ is a curve, we define its energy to be: $$ E(\gamma) = \frac{1}{2}\int_a^b \|\gamma'\|^2 $$ and one can show that, in the space of all curves $\{\gamma : [a,b] \to M\}$ with same end points, a curve $\gamma$ is a point where the energy functional is extremal if and only if $\nabla_{\gamma'}\gamma'=0$, that is if and only if $\gamma$ is a solution of the equation of geodesics. Hence, a minimizer of the energy functional is a geodesic.