Uniqueness of the torsion-curvature pair

TL;DR

This work extends a curvature-characterization result to arbitrary linear connections by classifying natural endomorphism-valued 2-forms and vector-valued 2-forms under Bianchi identities. Using the framework of natural operations and normal tensors, the authors show that the torsion and curvature form a unique (up to a scalar) natural pair satisfying both Bianchi identities. They prove that the space of closed natural endomorphism-valued 2-forms is 3-dimensional, spanned by $R$ and two trace-derived tensors, then deduce that any pair obeying the Bianchi constraints must be proportional to $(\mathrm{Tor}, R)$. The approach combines GL$(n)$-equivariant map analysis and computer-assisted verification, yielding a robust, general characterization with broad implications for natural geometric constructions.

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.

Theorems & Definitions (20)

• Definition 2.1
• Definition 2.2
• Proposition 2.3
• proof
• Proposition 2.4
• proof
• Definition 2.5
• Theorem 2.6
• Definition 2.7
• Definition 2.8