Uniqueness of the torsion-curvature pair

TL;DR

This paper proves that on smooth manifolds of dimension four or higher, the torsion and curvature forms are uniquely determined up to a scalar factor by their natural association with a linear connection, satisfying Bianchi identities.

Contribution

It extends a recent characterization of the curvature tensor of symmetric linear connections to arbitrary linear connections, establishing the uniqueness of torsion-curvature pairs under certain conditions.

Findings

Torsion and curvature are uniquely determined up to a scalar factor.

The result applies to manifolds of dimension four or higher.

It generalizes previous characterizations to all linear connections.

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.