A self-scaled barrier on a symmetric cone is a non-degenerate convex combination of the logarithms of the determinants of the irreducible factors of the cone. The special properties of the self-scaled barriers are at the heart of the interior-point methods used for solving conic optimization problems over symmetric cones. We introduce an analytic description of self-scaled barriers which is of local character and independent of the notion of a symmetric cone. Namely, we identify these barriers as the solutions of a certain quasi-linear fourth-order partial differential equation. Given such a solution in the neighbourhood of some point, it defines and can be extended to the interior of some symmetric cone on which it will represent a self-scaled barrier. This partial differential equation has a simple interpretation as the vanishing of a certain mixed covariant derivative of the metric defined by the Hessian of the solution with respect to the affine connection of the ambient real space and the Levi-Civita connection of the Riemannian metric defined by this Hessian. More precisely, the third derivative of the solution has to be invariant with respect to the geodesic flow defined by the Riemannian metric. Thus in a certain sense, self-scaled barriers resemble cubic polynomials.