Generalized Ricci and Bianchi identities for a connection with torsion non-tensor and curvature non-tensor
Abstract
A manifold is considered whose structure equations are constructed using a deformation of the exterior differential. The torsion and curvature objects of the affine connection on this manifold are not tensors. The curvature object is a tensor, and it is vanishing, only for a canonical connection. The torsion object coincides with the antisymmetry of fiber coordinates and it is non-vanishing even for the canonical connection. Unlike the torsion-free Levi-Civita connection, the canonical connection has vanishing curvature and non-vanishing torsion.
Generalized Ricci and Bianchi identities are constructed for curvature and torsion of the affine connection on this manifold. However, the repeated deformed differential for the basis forms and connection forms vanishes only along a line on the manifold. For the canonical connection, these identities take on a classical form. Moreover, in this case, the repeated deformed differential for the connection forms is identically equal to zero, and the repeated deformed differential for the basis forms vanishes only along the line on the manifold.
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