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

Question

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?

Answer 1

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.

It is the basis of torsion based gravity models.