I've been looking into the geometric interpretation of the Ricci tensor, and the standard idea is that, for example in 3D, $Ricci(u,u)$ corresponds to the rate of change of volumes in the direction u due to curvature.
Now, as alluded to in the answer to this question, $Ricci(u,v)$, where $u$ and $v$ are distinct vectors, can be solved for using the polarization identity, i.e.:
$Ricci(u+v,u+v) = Ricci(u,u) + Ricci(v,v) + 2{\cdot}Ricci(u,v)$.
So you could say that $2{\cdot}Ricci(u,v)$ takes care of the extra volume growth along the vector $u + v$ that can't be obtained by merely summing $Ricci(u,u)$ and $Ricci(v,v)$.
Unfortunately, at the moment, this means very little to me geometrically.
Is there a more direct, concrete, and less hand-wavey way to understand $Ricci(u,v)$?
Thanks in advance!