The role of torsion and a scalar field φ in gravitation, especially, in the presence of a Dirac field in the background of a particular class of the Riemann-Cartan geometry is considered here. Recently, a Lagrangian density with Lagrange multipliers has been proposed by the author which has been obtained by picking some particular terms from the SO(4, 1) Pontryagin density, where the scalar field φ causes the de Sitter connection to have the proper dimension of a gauge field. In this article the scalar field has been linked to the dimension of the Dirac field. Here we get the field equations for the Dirac field and the scalar field in such a way that both of them appear to be mutually non-interacting. In this scenario the scalar field appears to be a natural candidate for the dark matter and the dark radiation.

In the Einstein-Cartan space U 4 , an axial vector torsion together with a scalar field connected to a local scale factor have been considered. By combining two particular terms from the SO(4, 1) Pontryagin density and then modifying it in a SO(3, 1) invariant way, we get a Lagrangian density with Lagrange multipliers. Then under FRW-cosmological background, where the scalar field is connected to the source of gravitation, the Euler-Lagrange equations ultimately give the constancy of the gravitational constant together with only three kinds of energy densities representing mass, radiation and cosmological constant. The gravitational constant has been found to be linked with the geometrical Nieh-Yan density.

Riemann–Cartan space–time U4 is considered here. It has been shown that when we link topological Nieh–Yan density with the gravitational constant, we then obtain Einstein–Hilbert Lagrangian as a consequence.

Axial vector torsion in the Einstein-Cartan space U 4 is considered here. By picking a particular term from the SO(4, 1) Pontryagin density and then modifying it in a SO(3, 1) invariant way, we get a Lagrangian density with Lagrange multipliers. Then considering torsion and torsion-less connection as independent fields, it has been found that κ and λ of Einstein-Hilbert Lagrangian, appear as integration constants in such a way that κ has been found to be linked with the topological Nieh-Yan density of U 4 space.

In the gravity without metric formalism of Capovilla, Jacobson and Dell, the topological θ-term appears through a canonical transformation.The origin of this canonical transformation is probed here. It is shown here that when θ-term appears cosmological λ-term also appears simultaneously.PACS numbers : 04.20.Cv, 04.20.Fy Key words : Canonical transformation, Torsion, θ-term, Cosmological constant Einstein originally formulated the theory of gravitation as a set of differential equations obeyed by the metric tensor of space-time. Ashtekar 1,2 has rewritten Einstein's theory, in its hamiltonian formulation as a set of differential equations obeyed by an SO(3) connection and its canonically conjugate momentum. A Lagrangian formulation, in which the variables are the space-time tetrads and the self-dual spin connection, was given by Samuel 3 , Jacobson and Smolin 4 .If tetrad is invertible, one may eliminate the spin connection to obtain the conventional Hilbert action, plus a (complex) surface term, so that the Lagrangian formulation is given entirely in terms of the tetrad. However, it is clearly natural to enquire whether one can give a Lagrangian formulation of Ashtekar's theory in which the metric or tetrad, has been completely eliminated in favour of the connection. This has been done by Capovilla, Jacobson and Dell 5,6 .Bengtsson and Peldan 7 have shown that if one performs a particular canonical transformation involving Ashtekar variables and corresponding SO(3) gauge fields, the expression for the Hamiltonian constraint changes when other constraints remain unaffected. This corresponds precisely to the addition of a "CP-violating" θ-term to the CJD Lagrangian. Mullick and Bandyopadhyay 8 have shown that this "CPviolating" θ-term is responsible for non-zero torsion. This θ-term effectively corresponds to the chiral anomaly when a fermion chiral current interacts with a gauge field. In a recent paper 9 , it has been shown that chiral anomaly gives rise to the mass of a fermion which implies that for a massive fermion the divergence of the axial vector current (which is associated with torsion via θ-term) is non-vanishing.

In the torsion⊗curvature approach of gravity Chern-Simons modification has been considered here. It has been found that Chern-Simons contribution to the bianchi identity has become cancelled from that of the scalar field part. But "homogeneity and isotropy" consideration of present day cosmology is a consequence of the "strong equivalence principle" and vice-versa.1 It is to be noted here that in the presence of spinorial matter only the axial vector part of the torsion couples to the spinor field [8,9].

Starting from the multiplicative torsion approach of gravity and assuming a Killing vector to be proportional to the axial-vector matter current, here we derive Newton's law of gravity where the logarithm of the proportionality factor has been found to be the potential function.

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