What is the geometrical interpretation of Ricci tensor? In differential geometry and general relativity space is said to be flat if the Riemann tensor $R=0$. If the Ricci tensor on manifold $M$ is zero, it doesn't mean that the manifold itself is flat. So what's the geometrical meaning of Ricci tensor since it's been defined with the Riemann tensor as 
$$\mathrm{Ric}_{ij}=\sum_a R^a_{iaj}?$$ 
 A: The local geometric structure of a pseudo-Riemannian manifiold $M$ is completely described by the Riemann tensor $R_{abcd}$. The local structure of a manifold is affected by two possible sources 


*

*Matter sources in $M$: The matter distribution on a manifold is described by the stress tensor $T_{ab}$. By Einstein's equations, this can be related to the Ricci tensor (which is the trace of the Riemann tensor = $R_{ab} = R^c{}_{acb}$. 
$$
R_{ab} = 8 \pi G \left( T_{ab} + \frac{g_{ab} T}{2-d}  \right)
$$

*Gravitational waves on $M$. This is described by the Weyl tensor $C_{abcd}$ which is the trace-free part of the Riemann tensor. 
Thus, the local structure of $M$ is completely described by two tensors


*

*$R_{ab}$: This is related to the matter distribution. If one includes a cosmological constant, this tensor comprises the information of both matter and curvature due to the cosmological constant. 

*$C_{abcd}$: This describes gravitational waves in $M$. A study of Weyl tensor is required when describing quantum gravity theories. 
A: I've always liked the interpretation you get from the Raychaudhuri equation.  It shows you that the Ricci tensor tends to cause geodesics to focus together.  If you begin with a family of geodesics with tangent vector $u^a$, you can define the expansion $\theta\equiv \nabla_a u^a$ which measures the rate at which geodesics are spreading out or converging together.  As you move along a an integral curve of $u^a$, the Raychaudhuri equation tells you how the expansion changes as a function of curve's parameter, $\lambda$:
$$ \frac{d}{d\lambda}\theta = -\frac13\theta^2-\sigma_{ab}\sigma^{ab}+\omega_{ab}\omega^{ab}-R_{ab}u^au^b.$$
$\sigma_{ab}$ is called the shear and is related to the tendency of a cross section of the curves to distort toward and ellipsoid, and $\omega_{ab}$ is the vorticity and describes how the curves twist around each other.  The Ricci tensor appears in this equation with a minus sign, so that when $R_{ab}u^au^b$ is positive, it tends to decrease the expansion, which describes focusing of the geodesics.  
