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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?

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

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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.

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