  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.
