On an integrable geometrical foundation of gravity

In a talk at the conference {\it Geometrical Foundations of Gravity at Tartu 2017}, it was suggested that the affine spacetime connection could be associated with purely fictitious forces. This leads to gravitation in a flat and smooth geometry. Fermions are found to nevertheless couple with the metrical connection and a phase gauge field. The theory is reviewed in this proceeding, in a Palatini and in a metric-affine gauge formulation.