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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.

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.

Summary. -The Lagrangian formulation of classical field theories and in particular general relativity leads to a coordinate-free, fully covariant analysis of these constrained systems. This paper applies multisymplectic techniques to obtain the analysis of Palatini and self-dual gravity theories as constrained systems, which have been studied so far in the Hamiltonian formalism. The constraint equations are derived while paying attention to boundary terms, and the Hamiltonian constraint turns out to be linear in the multimomenta. The equivalence with Ashtekar's formalism is also established. The whole constraint analysis, however, remains covariant in that the multimomentum map is evaluated on any spacelike hypersurface. This study is motivated by the non-perturbative quantization program of general relativity. More recently, the work by Ashtekar, Rovelli, Smolin and their collaborators on connection dynamics and loop variables has made it possible to cast the constraint equations of general relativity in polynomial form, and then find a large class of solutions to the quantum version of constraints [7][8][9][10][11][12][13][14]. However, the quantum theory via the Rovelli-Smolin transform still suffers from severe mathematical problems in 3+1 space-time dimensions [15], and there appear to be reasons for studying non-perturbative quantum gravity also from a Lagrangian, rather than Hamiltonian, point of view (see below). The aim of this paper is therefore to provide a multisymplectic, Lagrangian framework for general relativity [16][17][18], to complement the present attempts to quantize general relativity in a non-perturbative way. The motivations of our analysis are as follows.(i) In the case of field theories, there is not a unique prescription for taking duals, on passing to the Hamiltonian formalism. For example, algebraic and topological duals are different. In turn, this may lead to inequivalent quantum theories. SPACE-TIME COVARIANT FORM OF ASHTEKAR'S CONSTRAINTS(ii) The 3+1 split of the Lorentzian space-time geometry, with the corresponding Σ×R topology, appears to violate the manifestly covariant nature of general relativity, as well as rely on a very restrictive assumption on the topology [19].(iii) In the Lagrangian formalism, explicit covariance is instead recovered. The first constraints one actually evaluates correspond to the secondary first-class constraints of the Hamiltonian formalism. At least at a classical level, the Lagrangian theory of constrained systems is by now a rich branch of modern mathematical physics [4,20,21], although the majority of general relativists are more familiar with the Hamiltonian framework.(iv) In the ADM formalism [5,6], the invariance group is not the whole diffeomorphism group, but a subgroup given by the Cartesian product of diffeomorphisms on the real line with the diffeomorphism group on spacelike three-surfaces. By contrast, in the Lagrangian approach, the invariance group of the theory is the full diffeomorphism group of fourdimensional Lorentzian space-time...

The small algebra of loop functionals, defined by Rovelli and Smolin, on the Ashtekar phase space of general relativity is studied. Regarded as coordinates on the phase space, the loop functionals become degenerate at certain points. All the degenerate points are found and the corresponding degeneracy is discussed. The intersection of the set of degenerate points with the real slice of the constraint surface is shown to correspond precisely the Goldberg-Kerr solutions. The evolution of the holonomy group of Ashtekar's connection is examined, and the complexification of the holonomy group is shown to be preserved under it. Thus, an observable of the gravitational field is constructed.

PACS. 14.60Pq -Neutrino mass and mixing. PACS. 95.30Sf -Relativity and gravitation. PACS. 04.50+h -Alternative theories of gravity.Abstract. -We show that in theories of gravitation with torsion the helicity of fermion particles is not conserved and we calculate the probability of spin flip, which is related to the anti-symmetric part of affine connection. Some cosmological consequences are discussed.Introduction. -Attempts to conciliate General Relativity with Quantum Theory yielded to propose theories of gravitation including torsion fields, so that the natural arena is the space-time U 4 that is a generalization of Riemann manifold V 4 .The advantage to pass from V 4 to U 4 is due to the fact that the spin of a particle turns out to be related to the torsion just as its mass is responsable of the curvature. From this point of view, such a generalization tries to include the spin fields of matter into the same geometrical scheme of General Relativity.One of the attempts in this direction is the Einstein-Cartan-Sciama-Kibble (ECSK) theory [1]. However the torsion seems to play an important role in any fundamental theory. For instance: a torsion field appears in (super)string theory if we consider string fundamental modes; we need, at least, a scalar mode and two tensor modes: a symmetric and antisymmetric one. The latter, in the low energy limit for string effective action, gives the effects of a torsion field [2]; any attempts of unification between gravity and electromagnetism require the inclusion torsion in four and in higher-dimensional theories as Kaluza-Klein ones [3]; theories of gravity formulated in terms of twistors need the inclusion of torsion [4]; in the supergravity theory torsion, curvature and matter fields are treated under the same standard [5]; in cosmology torsion could have had a relevant role into dynamics of the early universe because it gives a

We study the dynamics of a flat Friedmann-Robertson-Walker universe filled with a self-interacting scalar field nonminimally coupled to the gravitational field. Dynamical equations for the system can be derived from a pointlike Lagrangian. For this system an additional Nother symmetry exists provided that the coupling constant 6 is equal to 0 or i. When l= $ the scalar potential has to be constant. In this case we obtain an exact solution. We also analyze the behavior of the scalar field when S f 0, i .Most of the considered solutions are unphysical but there exists a very interesting case in which the effective cosmological constant is rapidly changing, which might lead to inflation.