  Let $U \subset \mathbb A^n$ be an open subset of real affine space. We
consider functions $F: U \to \mathbb R$ with non-degenerate Hessian such that
the first or the third derivative of $F$ is parallel with respect to the
Levi-Civita connection defined by the Hessian metric $F"$. In the former case
the solutions are given precisely by the logarithmically homogeneous functions,
while the latter case is closely linked to metrised Jordan algebras. Both
conditions together are related to unital metrised Jordan algebras. Both
conditions combined with convexity provide a local characterization of
canonical barriers on symmetric cones.
