Carlo Alberto Mantica

Weakly Z-symmetric manifolds

Abstract.We introduce a new kind of Riemannian manifold that includes weakly-, pseudo- and pseudo projective Ricci symmetric manifolds. The manifold is defined through a generalization of the so called Z tensor; it is named weaklyZ-symmetric and is denoted by (WZS)n. If the Z tensor is singular we give conditions for the existence of a proper concircular vector. For non singular Z tensors, we study the closedness property of the associated covectors and give sufficient conditions for the existence of a proper concircular vector in the conformally harmonic case, and the general form of the Ricci tensor. For conformally flat (WZS)n manifolds, we derive the local form of the metric tensor.

PSEUDO Z SYMMETRIC RIEMANNIAN MANIFOLDS WITH HARMONIC CURVATURE TENSORS

In this paper we introduce a new notion of Z-tensor and a new kind of Riemannian manifold that generalize the concept of both pseudo Ricci symmetric manifold and pseudo projective Ricci symmetric manifold. Here the Z-tensor is a general notion of the Einstein gravitational tensor in General Relativity. Such a new class of manifolds with Z-tensor is named pseudoZ symmetric manifold and denoted by (PZS)n. Various properties of such an n-dimensional manifold are studied, especially focusing the cases with harmonic curvature tensors giving the conditions of closeness of the associated one-form. We study (PZS)n manifolds with harmonic conformal and quasi-conformal curvature tensor. We also show the closeness of the associated 1-form when the (PZS)n manifold becomes pseudo Ricci symmetric in the sense of Deszcz (see [A. Derdzinsky and C. L. Shen, Codazzi tensor fields, curvature and Pontryagin forms, Proc. London Math. Soc.47(3) (1983) 15–26; R. Deszcz, On pseudo symmetric spaces, Bull. Soc. Math. Belg. Ser. A44 (1992) 1–34]). Finally, we study some properties of (PZS)4 spacetime manifolds.

A second-order identity for the Riemann tensor and applications

A second-order differential identity for the Riemann tensor is obtained, on a manifold with a symmetric connection. Several old and some new differential identities for the Riemann and Ricci tensors are derived from it. Applications to manifolds with recurrent or symmetric structures are discussed. The new structure of K-recurrency naturally emerges from an invariance property of an old identity due to Lovelock.

Pseudo-Z symmetric space-times

In this paper, we investigate Pseudo-Z symmetric space-time manifolds. First, we deal with elementary properties showing that the associated form Ak is closed: in the case the Ricci tensor results to be Weyl compatible. This notion was recently introduced by one of the present authors. The consequences of the Weyl compatibility on the magnetic part of the Weyl tensor are pointed out. This determines the Petrov types of such space times. Finally, we investigate some interesting properties of (PZS)4 space-time; in particular, we take into consideration perfect fluid and scalar field space-time, and interesting properties are pointed out, including the Petrov classification. In the case of scalar field space-time, it is shown that the scalar field satisfies a generalized eikonal equation. Further, it is shown that the integral curves of the gradient field are geodesics. A classical method to find a general integral is presented.

Weyl compatible tensors

We introduce the new algebraic property of Weyl compatibility for symmetric tensors and vectors. It is strictly related to Riemann compatibility, which generalizes the Codazzi condition while preserving much of its geometric implications. In particular, it is shown that the existence of a Weyl compatible vector implies that the Weyl tensor is algebraically special, and it is a necessary and sufficient condition for the magnetic part to vanish. Some theorems (Derdzinski and Shen [11], Hall [15]) are extended to the broader hypothesis of Weyl or Riemann compatibility. Weyl compatibility includes conditions that were investigated in the literature of general relativity (as in McIntosh et al. [16, 17]). A simple example of Weyl compatible tensor is the Ricci tensor of an hypersurface in a manifold with constant curvature.

RECURRENT Z FORMS ON RIEMANNIAN AND KAEHLER MANIFOLDS

In this paper, we introduce a new kind of Riemannian manifold that generalize the concept of weakly Z-symmetric and pseudo-Z-symmetric manifolds. First a Z form associated to the Z tensor is defined. Then the notion of Z recurrent form is introduced. We take into consideration Riemannian manifolds in which the Z form is recurrent. This kind of manifold is named (ZRF)n. The main result of the paper is that the closedness property of the associated covector is achieved also for rank(Zkl) > 2. Thus the existence of a proper concircular vector in the conformally harmonic case and the form of the Ricci tensor are confirmed for(ZRF)n manifolds with rank(Zkl) > 2. This includes and enlarges the corresponding results already proven for pseudo-Z-symmetric (PZS)n and weakly Z-symmetric manifolds (WZS)n in the case of non-singular Z tensor. In the last sections we study special conformally flat (ZRF)n and give a brief account of Z recurrent forms on Kaehler manifolds.

Generalized Robertson–Walker spacetimes — A survey

Generalized Robertson–Walker spacetimes extend the notion of Robertson–Walker spacetimes, by allowing for spatial non-homogeneity. A survey is presented, with main focus on Chens characterization ...

A condition for a perfect-fluid space-time to be a generalized Robertson-Walker space-time

A perfect-fluid space-time of dimension n>3 with 1) irrotational velocity vector field, 2) null divergence of the Weyl tensor, is a generalised Robertson-Walker space-time with Einstein fiber. Condition 1) is verified whenever pressure and energy density are related by an equation of state. The contraction of the Weyl tensor with the velocity vector field is zero. Conversely, a generalized Robertson-Walker space-time with null divergence of the Weyl tensor is a perfect-fluid space-time.

PSEUDO-Q-SYMMETRIC RIEMANNIAN MANIFOLDS

In this paper, we introduce a new kind of tensor whose trace is the well-known Z tensor defined by the present authors. This is named Q tensor: the displayed properties of such tensor are investigated. A new kind of Riemannian manifold that embraces both pseudo-symmetric manifolds (PS)n and pseudo-concircular symmetric manifolds is defined. This is named pseudo-Q-symmetric and denoted with (PQS)n. Various properties of such an n-dimensional manifold are studied: the case in which the associated covector takes the concircular form is of particular importance resulting in a pseudo-symmetric manifold in the sense of Deszcz [On pseudo-symmetric spaces, Bull. Soc. Math. Belgian Ser. A44 (1992) 1–34]. It turns out that in this case the Ricci tensor is Weyl compatible, a concept enlarging the classical Derdzinski–Shen theorem about Codazzi tensors. Moreover, it is shown that a conformally flat (PQS)n manifold admits a proper concircular vector and the local form of the metric tensor is given. The last section is devoted to the study of (PQS)n space-time manifolds; in particular we take into consideration perfect fluid space-times and provide a state equation. The consequences of the Weyl compatibility on the electric and magnetic part of the Weyl tensor are pointed out. Finally a (PQS)n scalar field space-time is considered, and interesting properties are pointed out.

The closedness of some generalized curvature 2-forms on a Riemannian manifold I

In this paper we study the closedness properties of generalized cur- vature 2-forms, which are said to be Riemannian, Conformal, Projective, Concircular and Conharmonic curvature 2-forms, associated to each generalized curvature tensors on a Riemannian manifold. Corresponding to each curvature tensors, such generalized curvature 2-forms are the associated curvature 2-forms. In particular, we focus on the closedness of difierential 2-forms associated to the divergence of generalized curvature tensors, which is weaker than the notion of harmonic curvature. In this case, we give an algebraic condition involving the Riemann curvature tensor and the Ricci tensor arising from an old identity due to Lovelock.