Ricci curvature tensor plays an important role in general relativity, where it is the key term in the Einstein field equations. It is known, the Ricci tensor defined by the Riemannian curvature tensor.
As I undestand, Ricci curvature is about difference between "an Euclidien ball" and "geodesic ball in manifold" in locally. If it vanish there will be no difference. This kind of manifolds are called Ricci flat. Also these manifolds are special case of Einstein manifolds (wiki). A physicist my friend told me that the Ricci curvature is important than Riemann curvature in physics.
Yes, Ricci curvature is more important than Riemannian curvature in physics. In fact, the Ricci tensor is related to the matter content of the universe via Einstein's field equation in general relativity theory. It is the part of the curvature of spacetime that determines the degree to which matter will tend to converge or diverge in time.
It is known, the Ricci curvature tensor is the trace of the Riemannian curvature tensor. Ricci curvature plsys an important role in general relativity. In particular, it is the key term in the Eienstien field equation. Geometrically, Ricci flat means solving the Eienstien field equation of Riemannian manifold with vanishing cosmological constant. For this reason, I agree with Inan Unal and Bang-Yen Chen. Ricci curvature tensor is more important than Riemannian curvature tensor.
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