Spinorial matter in affine theory of gravity and the space problem

The purely affine theory of gravity possesses a canonical formulation. For this and other reasons, it could be a promising candidate for quantum gravi ty. Motivated by these perspectives, we discuss spinorial matter coupled to gravity, where the. latter is described by a connection having no a priori relation to a metric. We show that one can establish a truncated spinor fo rmalism which, for special or approximate solutions to the gravitational eq uations, reduces to the standard formalism. As a consequence, one arrives a t "matter-induced" Riemann-Cartan spaces solving the Weyl-Cartan space prob lem.