the torsion-free Levi-Civita connection: Then we have that the full metric compatible affine connection can now be written as: Relationship to teleparallelism In the theory of teleparallelism, one encounters a connection, the Weitzenböck connection, which is flat (vanishing Riemann curvature) but has a non-vanishing torsion. The flatness is exactly what allows parallel frame fields to be constructed. These notions can be extended to supermanifolds. See also Belinfante–Rosenfeld stress–energy tensor References Tensors Riemannian geometry Connection (mathematics)