Riccitensor

The Ricci tensor, denoted Ric_{μν}, is a fundamental object in differential geometry and general relativity. It arises from contracting the Riemann curvature tensor: Ric_{μν} = R^λ_{ μ λ ν} = g^{αβ} R_{α μ β ν}. It is a symmetric (0,2)-tensor for a torsion-free (Levi-Civita) connection, meaning Ric_{μν} = Ric_{ν μ}.

The Ricci tensor encodes the way volumes change under parallel transport and contributes to the focusing of geodesic congruences. It is related to the scalar curvature R by a trace relation: R = g^{μν} Ric_{μν}. In two dimensions, Ric_{μν} is proportional to the metric: Ric_{μν} = (1/2) R g_{μν}, but in higher dimensions this is no longer generally true.

In local coordinates, the Ricci tensor can be computed from the Christoffel symbols Γ^λ_{ μ ν} via Ric_{μν} = ∂_λ Γ^λ_{ μ ν} - ∂_ν Γ^λ_{ μ λ} + Γ^λ_{ μ ν} Γ^σ_{ λ σ} - Γ^σ_{ μ λ} Γ^λ_{ ν σ}. It is a contraction of the full Riemann curvature tensor and thus contains less information than Riemann curvature, yet it plays a central role in describing curvature in many geometric and physical contexts.

In general relativity, the Einstein tensor G_{μν} = Ric_{μν} - (1/2) g_{μν} R combines the Ricci tensor with the scalar curvature and appears in the Einstein field equations: G_{μν} = 8πG T_{μν} (ignoring a possible cosmological constant). In vacuum spacetimes, Ric_{μν} = 0.