The stress-energy tensor is associated to a volume density and flux in 4-spacetime and the Einstein tensor seems to represent a three-dimensional curvature (being a one-form with vector values) that acts on and is acted by the stress-energy source. If this is correct, I'm not sure what is this 3-curvature associated to in the most accepted cosmological model (L-CDM) that has no spatial curvature. Another question is referred to the geometrical meaning of the Einstein tensor, when it is said that it is an average of the Riemann surface curvature (a tensor(1,1)-valued 2-form after raising an index with the inverse metric tensor) over planes, what does it mean exactly? I think that you can obtain the Einstein tensor by applying the delta Kronecker to the Riemann 2-form to make a some kind of isotropic average tensor but I'm not sure how to do it. Also what is the intuitive geometrical difference between the Ricci and Einstein curvature? I think the Ricci curvature is more related to an average of sectional curvatures (and is also usually defined as the deviation a geodesic ball in curved spacetime has from the euclidean standard ball).