Gauge Gravitation Theory. What is the Geometry of the World?
  When joined the unified gauge picture of fundamental interactions, the gravitation theory leads to geometry of a space-time which is far from simplicity of pseudo-Riemannian geometry of Einstein's General Relativity. This is geometry of the affine-metric composite dislocated manifolds. The goal is modification of the familiar equations of a gravitational field and entirely the new equations of its deviations. In the present brief, we do not detail the mathematics, but discuss the reasons why it is just this geometry. The major physical underlying reason lies in spontaneous symmetry breaking when the fermion matter admits only the Lorentz subgroup of world symmetries of the geometric arena. 
G.Sardanashvily