Geometric Massive Gravity in Multi-Connection Framework

What is the right way to interpret a massive graviton? We generalize the kinematical framework of general relativity to multiple connections. The average of the connections is itself a connection and plays the role of the canonical connection in standard General Relativity. At the level of dynamics, the simplest choice of the Einstein-Hilbert action is indistinguishable from the single-connection case. However, inspired by Weyl geometry, we show how one can construct massive gravity to all orders in perturbation theory compatible with the de Rham-Gabadadze-Tolley ghost-free model. We conclude that the mass of the graviton can be interpreted as a geometrical property of spacetime arising from two connections. Furthermore in the multi-connection framework there is no ambiguity in the definition of physical metric and consequently coupling to matter.