# Is the Weitzenböck connection the only connection with torsion, but without curvature?

In teleparallel gravity, the (local) connection coefficients of the Weitzenböck connection are given by

$$\Pi^{\beta}{}_{\mu\nu}= h^{\beta}_{i} \partial_{\nu}h^{i}_{\mu} - \Gamma^{\beta}{}_{\mu\nu} \,$$

where $\Gamma^{\beta}{}_{\mu\nu}$ is the Levi-Civita connection.

The question is: Is there another connetion without curvature but with torsion, where the torsion is related to the curvature of the Levi-Civita connecion?

• It seems there are, check this recent paper, where they study Deformed Weitzenböck Connections in the context of Teleparallel Gravity. Aug 3, 2017 at 16:44

Not really, the Weitzenböck connection is what relates torsion without having to involve curvature. Even more exotic types of theories such as $F(T,T)$ or $F(T,G)$ gravity use the Weitzenböck connection for calculations.