Why do we must know the Weyl tensor for 4-dimensional space-time? I heard that we must know the Weyl tensor for fully describing the curvature of the 4-dimensional space-time (in space-time with less dimensions it vanishes, so I don't interesting in cases of less dimensions). So I have the question: what is physical (or geometrical) sense of the Weyl tensor and why don't we need only Riemann tensor for describing the curvature? Does it connected with gravitational waves directly?
 A: The Riemann tensor encapsulates all information about the 4-dimensional space-time. This information can generally divided into two sectors:


*

*Information about the curvature of space-time due to the existence of matter. This is given by the Ricci tensor according to the Einstein equation
$$
R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R = 8 \pi G T_{\mu\nu}
$$

*Information about the structure of gravitational waves in the space-time. This is given by the trace-free part of the Riemann  tensor, namely the Weyl tensor. Often, we are not quite interested in the exact structure of the space-time, but only if gravitational waves can exist or their structure. In these cases, one studies the Weyl tensor rather than the Ricci tensor. For example, in the setup of quantum gravity, one requires to study the asymptotic structure of spacetime. In these theories, a good understanding of the Weyl tensor is more important. 
A: The Weyl tensor contains the information necessary to describe solutions of the Einstein equations in vacuum, given by
$$R_{\mu\nu}=0.$$
From this we can deduce that the trace part of the Riemann tensor vanishes, but not its traceless part, which is given by the Weyl tensor. The latter therefore describes curvature phenomena in the absence of matter, like gravitational waves. 
