Gravity and induced matter on nearly Kähler manifolds

Abstract

We show that the conservation of energy-momentum tensor of a gravitational model with Einstein-Hilbert like action on a nearly Kähler manifold with the scalar curvature of a curvature-like tensor, is consistent with the nearly Kähler properties. In this way, the nearly Kähler structure is automatically manifested in the action as a induced matter field. As an example of nearly Kähler manifold, we consider the group manifold of R × II × S 3 × S 3 on which we identify a nearly Kähler structure and solve the Einstein equations to interpret the model. It is shown that the nearly Kähler structure in this example is capable of alleviating the well known fine tuning problem of the cosmological constant. Moreover, this structure may be considered as a potential candidate for dark energy. * 1 The complex metric is based on Einstein's attempts to generalize the relativistic theory of gravitation to establish a unified field theory [3,4].

String gravitational equations with Hermitian structure

“…In this paper, we are interested in the idea of our recent paper to establish a physical interpretation for the complex structures [16]. We know that in general relativity the gravitational effects are characterized by a geometrical structure, so called metric, on a Riemannian manifold.…”

“…But, on a complex manifold there is an additional geometrical structure, namely the complex structure J, which may be capable of carrying physical significance. In this regard, we have already presented a matter interpretation for the almost complex structure [16]. Here, following the same idea we will investigate a σ-model where the role of antisymmetric B-field tensor is played by the complex structure (or more clearly by the fundamental antisymmetric Kähler form Ω which is related to the complex structure with a hermitian metric).…”