f(R)-gravity with torsion: a geometric approach within the J-bundles framework

We discuss the $f(R)$-theories of gravity with torsion in the framework of ${\cal J}$-bundles. Such an approach is particularly useful since the components of the torsion and curvature tensors can be chosen as fiber ${\cal J}$-coordinates on the bundles and then the symmetries and the conservation laws of the theory can be easily achieved. Field equations of $f(R)$-gravity are studied in empty space and in presence of various forms of matter as Dirac fields, Yang--Mills fields and spin perfect fluid. Such fields enlarge the jet bundles framework and characterize the dynamics. Finally we give some cosmological applications and discuss the relations between $f(R)$-gravity and scalar-tensor theories.