Prescribing geodesics and a variational problem for Riemannian metrics

TL;DR

This paper introduces a variational framework to determine Riemannian metrics whose geodesics match prescribed unparametrised paths, with special results in the conformal case and applications to convex projective surfaces.

Contribution

It defines a new functional on Riemannian metrics that vanishes precisely when geodesics match prescribed paths and analyzes its conformal critical points, including existence and uniqueness results.

Findings

The functional $\\mathcal{E}$ vanishes iff geodesics match prescribed paths.

Conformal critical points satisfy a Yamabe-type equation.

Every conformal class on a surface contains a conformally critical metric.

Abstract

Given a prescription of unparametrised paths on a manifold $M$, one path for each tangent direction, we may ask whether these paths agree with the geodesics of a Riemannian metric on $M$. Generically, this is not the case. Motivated by this fact, we introduce a non-negative functional $\mathcal{E}$ on the space of Riemannian metrics on $M$ so that $\mathcal{E}(g)=0$ if and only if the geodesics of the metric $g$ agree with the prescribed paths. We compute the variational equations for $\mathcal{E}$ and show that the conformal variational equation is, perhaps surprisingly, of Yamabe type. This allows us to obtain existence results for conformally critical points of $\mathcal{E}$. In particular, in the surface case, every conformal class contains a conformally critical metric, unique up to homothety. As a by-product, we establish that the Blaschke metric of a properly convex projective surface is a critical point for $\mathcal{E}$.