Uniqueness of the Torsion–Curvature Pair

On smooth manifolds of dimension n ≥ 4 n ≥ 4 , we prove that the torsion and curvature are, up to a scalar factor, the only pair of a vector-valued 2-form and an endomorphism-valued 2-form naturally associated with a linear connection that satisfy both the linear and differential Bianchi identities. This result extends to arbitrary linear connections a recent characterization of the curvature tensor of a symmetric linear connection obtained in Gordillo-Merino and Martínez-Bohórquez (Rev R Acad Cienc Exact Fis Nat Ser A Mat RACSAM 118(7):17, 2024).