TITLE: Geometrical interpretation of Ricci curvature
QUESTION [58 upvotes]: I see the scalar curvature $R$ as an indicator of how a manifold curves locally (the easiest example is for a $2$-dimensional manifold $M$, where the $R=0$ in a point means that it is flat there, $R>0$ that it makes like a hill and $R<0$ that it is a saddle point).
Are there analogous interpretations for the Riemann tensor $Rm$ and for the Ricci curvature $Rc$? I tried to think about it but I really can't get anything making sense.

REPLY [8 votes]: In this context, most of the "interpretations" are useless.
The main question is where the curvature shows up.
For Ricci curvature there are three places:

Bochner's formula for 1-forms.
First variation for the mean curvature of hypersurface.
Ricci flow

Everything known comes from these, one way or an other.