Bianchi I (algebraic): R^ρ_{[σμν]}=0; II (differential): ∇_{[λ}R_{|σ|μν]}=0

Two Bianchi identities of the Levi-Civita connection. First ('algebraic') Bianchi identity: the cyclic sum R^ρ_{σμν} + R^ρ_{μνσ} + R^ρ_{νσμ} = 0 — equivalently R^ρ_{[σμν]} = 0 in anti-symmetrisation notation. Proof via torsion-freeness:…