The role of torsion in f (R) gravity is considered in the framework of metricaffine formalism. We discuss the field equations in empty space and in the presence of perfect fluid matter, taking into account the analogy with the Palatini formalism. As a result, the extra curvature and torsion degrees of freedom can be dealt as an effective scalar field of a fully geometric origin. From a cosmological point of view, such a geometric description could account for the whole dark side of the universe.

f (R)-gravity with geometric torsion (not related to any spin fluid) is considered in a cosmological context. We derive the field equations in vacuum and in presence of perfect-fluid matter and discuss the related cosmological models. Torsion vanishes in vacuum for almost all arbitrary functions f (R) leading to standard General Relativity. Only for f (R) = R 2 , torsion gives contribution in the vacuum leading to an accelerated behavior . When material sources are considered, we find that the torsion tensor is different from zero even with spinless material sources. This tensor is related to the logarithmic derivative of f ′ (R), which can be expressed also as a nonlinear function of the trace of the matter energy-momentum tensor Σµν . We show that the resulting equations for the metric can always be arranged to yield effective Einstein equations. When the homogeneous and isotropic cosmological models are considered, terms originated by torsion can lead to accelerated expansion. This means that, in f (R) gravity, torsion can be a geometric source for acceleration.

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We discuss the f (R)-theories of gravity with torsion in the framework of J -bundles. Such an approach is particularly useful since the components of the torsion and curvature tensors can be chosen as fiber 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.

We study Dirac spinors in Bianchi type-I cosmological models, within the framework of torsional $f(R)$-gravity. We find four types of results: the resulting dynamic behavior of the universe depends on the particular choice of function $f(R)$; some $f(R)$ models do not isotropize and have no Einstein limit, so that they have no physical significance, whereas for other $f(R)$ models isotropization and Einsteinization occur, and so they are physically acceptable, suggesting that phenomenological arguments may select $f(R)$ models that are physically meaningful; the singularity problem can be avoided, due to the presence of torsion; the general conservation laws holding for $f(R)$-gravity with torsion ensure the preservation of the Hamiltonian constraint, so proving that the initial value problem is well-formulated for these models.Comment: 25 pages, 1 figur

In connection with this, a section is said critical if ␦A L / ␦X͑͒ = 0 for all compact domains D and all deformations J constant on the boundary ‫ץ‬D. Due to this last condition at the boundary, it follows that a section is critical if and only if the equation 4456

In the gauge natural bundle framework a new space is introduced and a first-order purely frame-formulation of General Relativity is obtained. In some of our recent works [1, 2, 3] a new geometrical framework for YangMills field theories and General Relativity in the tetrad-affine formulation has been developed.The construction of the new geometrical setting has been obtained quotienting the first-jet bundles of the configuration spaces of the above theories in a suitable way, resulting into the introduction of a new family of fiber bundles.In this letter we show that these new spaces allow a (covariant) first-order purely frame-formulation of General Relativity.The whole geometrical construction will be developed within the gauge natural bundle framework [4], which provides the suitable mathematical setting for globally describing gravity in the tetrad formalism.To start with, let M be a space-time manifold, allowing a metric tensor g with signature η = (1, 3): the manifold M will be called a η-manifold and the metric tensor canonical representation will be η µν := diag (−1, 1, 1, 1). Moreover, let L(M ) be the frame-bundle over M and P → M a principal fiber bundle over M with structural group SO (1, 3).The configuration space of the theory (the tetrad space) is a GL(4, ℜ) bundle π : E → M , associated to P × M L(M ) through the left-action λ : (SO(1, 3) × GL(4, ℜ)) × GL(4, ℜ) → GL(4, ℜ), λ(Λ, J; X) = Λ · X · J −1 (1) 1

A gauge-independent formulation of the theory developed in J. Phys. A: Math. Gen. 36 8341 ( 2003) is proposed.