On the α-connections and the α-conformal equivalence on statistical manifolds

PurposeIn this paper, we give some properties of the α-connections on statistical manifolds and we study the α-conformal equivalence where we develop an expression of curvature R¯ for ∇¯ in relation to those for ∇ and ∇^.Design/methodology/approachIn the first section of this paper, we prove some results about the α-connections of a statistical manifold where we give some properties of the difference tensor K and we determine a relation between the curvature tensors; this relation is a generalization of the results obtained in [1]. In the second section, we introduce the notion of α-conformal equivalence of statistical manifolds treated in [1, 3], and we construct some examples.FindingsWe give some properties of the difference tensor K and we determine a relation between the curvature tensors; this relation is a generalization of the results obtained in [1]. In the second section, we introduce the notion of α-conformal equivalence of statistical manifolds, we give the relations between curvature tensors and we construct some examples.Originality/valueWe give some properties of the difference tensor K and we determine a relation between the curvature tensors; this relation is a generalization of the results obtained in [1]. In the second section, we introduce the notion of α-conformal equivalence of statistical manifolds, we give the relations between curvature tensors and we construct some examples.

Geometric phase of cosmological scalar and tensor perturbations in f(R) gravity

By using the conformal equivalence of f(R) gravity in vacuum and the usual Einstein theory with scalar-field matter, we derive the Hamiltonian of the linear cosmological scalar and tensor perturbations in f(R) gravity in the form of time-dependent harmonic oscillator Hamiltonians. We find the invariant operators of the resulting Hamiltonians and use their eigenstates to calculate the adiabatic Berry phase for sub-horizon modes as a Lewis–Riesenfeld phase.

York’s Solution to the Initial-Value Problem

In this chapter I briefly review York’s method (or the conformal method) for solving the initial value problem of (GR). This method, developed initially by Lichnerowicz and then generalized by Choquet-Bruhat and York, allows to find solutions of the constraints of (GR) (in particular the Hamiltonian, or refoliation constraint) by scanning the conformal equivalence class of spatial metrics for a solution of the Hamiltonian constraint, exploiting the fact that, in a particular foliation (CMC), the transverse nature of the momentum field is preserved under conformal transformations. This method allows to transform the initial value problem into an elliptic problem for the solution for which good existence and uniqueness theorems are available. Moreover this method allows to identify the reduced phase space of (GR) with the cotangent bundle to conformal superspace (the space of conformal 3-geometries), when the CMC foliation is valid. SD essentially amounts to taking this phase space as fundamental and renouncing the spacetime description when the CMC foliation is not available.

Historical Interlude

This chapter recounts the main steps that led to the present formulation of SD. In particular, the attempts by Barbour, Ó Murchadha, Anderson, Kelleher and Foster of implementing conformal symmetry through best matching, which culminated in the realization that York’s method (in particular, the assumption of not transforming the trace of the extrinsic curvature when scanning the conformal equivalence class in a CMC slice) can be understood as a volume-preserving conformal transformation.