Uniqueness of the Torsion–Curvature Pair

Key Points

• To demonstrate the uniqueness of the torsion and curvature pair associated with linear connections on smooth manifolds of dimension n ≥ 4.
• Proved relationships between torsion and curvature using mathematical analysis.
• Extended previous findings regarding symmetric linear connection curvature.
• Investigated linear and differential Bianchi identities in the context of manifold geometry.
• Established that torsion and curvature are uniquely defined up to a scalar factor.
• Confirmed these pairs satisfy the linear and differential Bianchi identities in higher-dimensional manifolds.
• Expanded a characterization previously held for symmetric linear connections to arbitrary linear connections.