Let $(M,g)$ be a Riemannian manifold of dimension $n>2$. Thanks to the late T.Branson we have the following definition of the so-called $Q$-curvature:

$Q= \Delta R + \frac{n^3-4n^2+16n-16}{4(n-1)(n-2)^2} R^2 - \frac{8(n-1)}{(n-2)^2}|Ric|^2.$

Here $\Delta = -div\nabla$, $R$ is the scalar curvature, and $|Ric|$ is the norm of the Ricci tensor. There has been much research on $Q$ curvature since its discovery in the eighties, motivated in large part by the conformal transformation properties that it possesses. A question that appears to be open, though, is whether or not there is a (relatively) concise geometric interpretation of this scalar curvature invariant. For $R$ we have the nice interpretation that it determines the rate at which the growth of a ball around a point differs from the flat case. Similarily the Ricci tensor measures the deviation of a solid angle from the Euclidean case. Can you think of a geometric interpretation of $Q$-curvature that is similarily elegant?