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A bi-connection modified gravity is considered in the presence of an additional Gauss-Bonnet term to standard Einstein-Hilbert action. The standard Einstein-Hilbert does not allow propagating additional degrees of freedom in bi-connection model. This fact makes the model exactly the same as standard general relativity with just one metric. In a general framework, we show the existence of Gauss-Bonnet term can solve this problem. Then we focus on bi-connection model in the Weyl framework which makes the analysis and the results more straightforward. In this framework there is a self-accelerating solution for the metric while the Weyl vector has a non-trivial dynamics. At last we show two independent realizations of de Rham-Gabadadze-Tolley (dRGT) massive gravity in the Weyl geometry framework. Though the potential term can mimic dRGT mass term but there is also mixing term between Riemann curvature tensor and the potential term.
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