Uniqueness of the torsion-curvature pair

Abstract

On smooth manifolds of dimension $n \ge 4$, 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 characterisation of the curvature tensor of a symmetric linear connection obtained in the paper "On the uniqueness of the torsion and curvature operators", Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM, 114, 2020.